The linear model

# The SPINE of statistics: the linear model, parameters and estimation
## Professor Andy Field

<div>
  <img style="vertical-align:middle; width:30px; height:30px" src="media/twitter_60.png">
  <span style="line-height:40px;">@profandyfield</span>
</div>
<div>
  <img style="vertical-align:middle; width:60px" src="media/youtube.png">
  <span style="line-height:40px;">www.youtube.com/user/ProfAndyField/</span>
</div>
<div>
  <img style="vertical-align:middle; width:30px; height:30px" src="media/ds_com_fav.png">
  <span style="line-height:40px;">www.discoveringstatistics.com</span>
</div>
<div>
  <img style="vertical-align:middle; width:30px; height:30px" src="media/milton_grey_fav.png">
  <span style="line-height:40px;">www.milton-the-cat.rocks</span>
</div>
<div>
  <img style="vertical-align:middle; width:30px; height:30px" src="media/discovr_fav.png">
  <span style="line-height:40px;">www.discovr.rocks</span>
</div>

???
    h or ?: Toggle the help window
    j: Jump to next slide
    k: Jump to previous slide
    b: Toggle blackout mode
    m: Toggle mirrored mode.
    p: Toggle PresenterMode
    f: Toggle Fullscreen
    t: Reset presentation timer
    <number> + <Return>: Jump to slide <number>
    c: Create a clone presentation on a new window

---

![:scale 83%](media/spine_content_map_base.png)
???

All research starts with a question, hypothesis, or something that needs to be quantified or estimated.

---
class: center

![:scale 83%](media/spine_content_map_se.png)

???

To test that idea you need to sample data. A key concept here is something known as the standard error.
---
class: center

![:scale 83%](media/spine_content_map_vis.png)

???

Having collected data you need to visualise it (although we don't talk about this in lectures, the tutorials will give you practice)
---
class: center

![:scale 83%](media/spine_content_map_glm.png)

???

You then fit a model to the data
---
class: center

![:scale 83%](media/spine_content_map_estimation.png)

???
When you fit a model there are typically two things you might be interested in. The first is estimating the effect of interest. You do this with a single value (a  parameter) or a range of values (and interval)

---
class: center

![:scale 83%](media/spine_content_map_nhst.png)

???
When you fit a model there are typically two things you might be interested in. The second is to test hypotheses about the parameters (usually whether they are different to 0)

---
class: center

![:scale 83%](media/spine_content_map_assump.png)

???
Having fit a model you want to test whether it is biased.
---

![:scale 83%](media/spine_content_map.png)
???
If it is biased you would want to fit one of a class of alternative models robust to these biases.
---

# The SPINE of statistics

## 5 Key concepts

* **S**tandard error

* **P**arameters

* **I**nterval estimates
 
* **N**ull hypothesis significance testing (NHST)

* **E**stimation

---
class: center

![:scale 83%](media/spine_lec_01.png)

???

Today we're going to look at the model we typically fit - the linear model - and look at how parameters are estimated (very broadly)

---
#  Learning outcomes

--
* Understand the commonalities in psychological statistical models
    + Most psychological statistical boil down to a very simple idea of predicting an outcome from one or more measured variables

--
* Understand the function and form of the linear model
  + Predicting an outcome variable from another variable (a predictor)
    + The mathematical model
    + Visualizing the model
    + Familiar models as variants of the linear model

--
* Understand what the model parameters (*b*s) represent

--
* Understand why we use sampling

--
* Understand (conceptually) least squares estimation

---
class: center, middle, title-slide, inverse
layout: false

## Why do (some) students hate statistics?

---
background-image: none
background-color: #000000

---
class: center, middle

.eq_lrge[
.ong_dk[
<p style="position:absolute; left: 10%; top: 15%;">
$$
Y_i = \hat{b}_0 + \hat{b}_1\text{X}_{i} + \text{e}_i
$$
</p>
]]

--
.eq_lrge[
.ong_dk[
<p style="position:absolute; left: 55%; top: 15%;">
$$
r = \frac{\sum\limits_{i = 1}^n(x_i-\bar{X})(y_i-\bar{Y})}{(N-1)s_xs_y}
$$
</p>
]]
--

.eq_lrge[
.ong_dk[
<p style="position:absolute; left: 10%; top: 55%;">
$$
t = \frac{\bar{X}_1-\bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}
$$
</p>
]]

.eq_lrge[
.ong_dk[
<p style="position:absolute; left: 55%; top: 55%;">
$$
\chi^2 = \sum\frac{(\text{observed}_{ij}-\text{model}_{ij})^2}{\text{model}_{ij}}
$$
</p>
]]

---
class: center, middle, title-slide, inverse
layout: false

## Can we make statistics (a bit) less like this?

---
<video width="100%" height="100%" controls id="my_video">
    <source src="media/andy_dungeon_scream.mp4" type="video/mp4">
</video>
---
layout: true
background-image: url("media/ais_ppt_hex_white.jpg")
background-size: cover
---
# The only equation you will ever need
## The General Linear Model (GLM)
--

.ong_dk[
$$
`\begin{aligned}
\text{outcome}_i &= (\text{model}_i) + \text{error}_i \\
\text{outcome}_i &= \hat{b}_0 + \hat{b}_1\text{predictor}_{i} + \dots + \text{error}_i
\end{aligned}`
$$
]

---
background-image: url("media/andy_zombie.jpg")
background-size: cover

# .bwn[A zombie quiz]
---
# A zombie quiz

> A researcher counted how many humans and zombies choose brain chips or potato chips to accompany their dinner at the university canteen

* How do I analyze these data?

<table class="table table-striped" style="margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> Organism </th>
   <th style="text-align:right;"> Brain chips </th>
   <th style="text-align:right;"> Potato chips </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Human </td>
   <td style="text-align:right;"> 28 </td>
   <td style="text-align:right;"> 42 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Zombie </td>
   <td style="text-align:right;"> 61 </td>
   <td style="text-align:right;"> 57 </td>
  </tr>
</tbody>
</table>
---
# A chi-square test

```r
chisq.test(zom_fct_tib$organism, zom_fct_tib$chip, correct = FALSE)
```

```
## 
## 	Pearson's Chi-squared test
## 
## data:  zom_fct_tib$organism and zom_fct_tib$chip
*## X-squared = 2.4105, df = 1, p-value = 0.1205
```

---
# A Spearman correlation?

```r
zom_tib |>
  correlation::correlation(method = "spearman")
```

<table class="table table-striped" style="margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> Parameter1 </th>
   <th style="text-align:left;"> Parameter2 </th>
   <th style="text-align:right;"> rho </th>
   <th style="text-align:right;"> CI </th>
   <th style="text-align:right;"> CI_low </th>
   <th style="text-align:right;"> CI_high </th>
   <th style="text-align:right;"> S </th>
   <th style="text-align:right;"> p </th>
   <th style="text-align:left;"> Method </th>
   <th style="text-align:right;"> n_Obs </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> organism </td>
   <td style="text-align:left;"> chip </td>
   <td style="text-align:right;"> -0.113 </td>
   <td style="text-align:right;"> 0.95 </td>
   <td style="text-align:right;"> -0.256 </td>
   <td style="text-align:right;"> 0.035 </td>
   <td style="text-align:right;"> 1232810 </td>
   <td style="text-align:right;background-color: yellow !important;"> 0.122 </td>
   <td style="text-align:left;"> Spearman correlation </td>
   <td style="text-align:right;"> 188 </td>
  </tr>
</tbody>
</table>

---
# A Kendall's `$\tau$` correlation?

```r
zom_tib |>
  correlation::correlation(method = "kendall")
```

<table class="table table-striped" style="margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> Parameter1 </th>
   <th style="text-align:left;"> Parameter2 </th>
   <th style="text-align:right;"> tau </th>
   <th style="text-align:right;"> CI </th>
   <th style="text-align:right;"> CI_low </th>
   <th style="text-align:right;"> CI_high </th>
   <th style="text-align:right;"> z </th>
   <th style="text-align:right;"> p </th>
   <th style="text-align:left;"> Method </th>
   <th style="text-align:right;"> n_Obs </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> organism </td>
   <td style="text-align:left;"> chip </td>
   <td style="text-align:right;"> -0.113 </td>
   <td style="text-align:right;"> 0.95 </td>
   <td style="text-align:right;"> -0.206 </td>
   <td style="text-align:right;"> -0.018 </td>
   <td style="text-align:right;"> -1.548 </td>
   <td style="text-align:right;background-color: yellow !important;"> 0.122 </td>
   <td style="text-align:left;"> Kendall correlation </td>
   <td style="text-align:right;"> 188 </td>
  </tr>
</tbody>
</table>

---
# A Pearson correlation?

```r
zom_tib |>
  correlation::correlation()
```

<table class="table table-striped" style="margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> Parameter1 </th>
   <th style="text-align:left;"> Parameter2 </th>
   <th style="text-align:right;"> r </th>
   <th style="text-align:right;"> CI </th>
   <th style="text-align:right;"> CI_low </th>
   <th style="text-align:right;"> CI_high </th>
   <th style="text-align:right;"> t </th>
   <th style="text-align:right;"> df_error </th>
   <th style="text-align:right;"> p </th>
   <th style="text-align:left;"> Method </th>
   <th style="text-align:right;"> n_Obs </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> organism </td>
   <td style="text-align:left;"> chip </td>
   <td style="text-align:right;"> -0.113 </td>
   <td style="text-align:right;"> 0.95 </td>
   <td style="text-align:right;"> -0.252 </td>
   <td style="text-align:right;"> 0.03 </td>
   <td style="text-align:right;"> -1.554 </td>
   <td style="text-align:right;"> 186 </td>
   <td style="text-align:right;background-color: yellow !important;"> 0.122 </td>
   <td style="text-align:left;"> Pearson correlation </td>
   <td style="text-align:right;"> 188 </td>
  </tr>
</tbody>
</table>

---
# A *t*-test?

```r
brains <- zom_tib |> dplyr::filter(chip == 0)
potatoes <- zom_tib |> dplyr::filter(chip == 1)
```

```r
t.test(brains$organism, potatoes$organism)
```

<table class="table table-striped" style="margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:right;"> estimate </th>
   <th style="text-align:right;"> estimate1 </th>
   <th style="text-align:right;"> estimate2 </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p.value </th>
   <th style="text-align:right;"> parameter </th>
   <th style="text-align:right;"> conf.low </th>
   <th style="text-align:right;"> conf.high </th>
   <th style="text-align:left;"> method </th>
   <th style="text-align:left;"> alternative </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 0.11 </td>
   <td style="text-align:right;"> 0.685 </td>
   <td style="text-align:right;"> 0.576 </td>
   <td style="text-align:right;"> 1.559 </td>
   <td style="text-align:right;background-color: yellow !important;"> 0.121 </td>
   <td style="text-align:right;"> 185.619 </td>
   <td style="text-align:right;"> -0.029 </td>
   <td style="text-align:right;"> 0.248 </td>
   <td style="text-align:left;"> Welch Two Sample t-test </td>
   <td style="text-align:left;"> two.sided </td>
  </tr>
</tbody>
</table>

---
# One-way ANOVA?

```r
zom_tib |>
  aov(organism ~ factor(chip), data = _)
```

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> df </th>
   <th style="text-align:right;"> sumsq </th>
   <th style="text-align:right;"> meansq </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p.value </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> factor(chip) </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0.563 </td>
   <td style="text-align:right;"> 0.563 </td>
   <td style="text-align:right;"> 2.416 </td>
   <td style="text-align:right;background-color: yellow !important;"> 0.122 </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: white !important;"> Residuals </td>
   <td style="text-align:right;background-color: white !important;"> 186 </td>
   <td style="text-align:right;background-color: white !important;"> 43.373 </td>
   <td style="text-align:right;background-color: white !important;"> 0.233 </td>
   <td style="text-align:right;background-color: white !important;"> NA </td>
   <td style="text-align:right;background-color: yellow !important;background-color: white !important;"> NA </td>
  </tr>
</tbody>
</table>

---
# Linear model (Regression)?

```r
zom_tib |>
  lm(organism ~ factor(chip), data = _)
```

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> estimate </th>
   <th style="text-align:right;"> std.error </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p.value </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;background-color: white !important;"> (Intercept) </td>
   <td style="text-align:right;background-color: white !important;"> 0.685 </td>
   <td style="text-align:right;background-color: white !important;"> 0.051 </td>
   <td style="text-align:right;background-color: white !important;"> 13.390 </td>
   <td style="text-align:right;background-color: yellow !important;background-color: white !important;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> factor(chip)1 </td>
   <td style="text-align:right;"> -0.110 </td>
   <td style="text-align:right;"> 0.071 </td>
   <td style="text-align:right;"> -1.554 </td>
   <td style="text-align:right;background-color: yellow !important;"> 0.122 </td>
  </tr>
</tbody>
</table>

---
# Loglinear model?

```r
xtabs(~ organism + chip, data = zom_fct_tib) |>
  MASS::loglm(~ organism + chip, data = _)
```

---
# Multilevel model?

```r
lmerTest::lmer(chip ~ organism + (1|canteen), data = zom_tib)
```

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Parameter </th>
   <th style="text-align:right;"> Coefficient </th>
   <th style="text-align:right;"> SE </th>
   <th style="text-align:right;"> CI </th>
   <th style="text-align:right;"> CI_low </th>
   <th style="text-align:right;"> CI_high </th>
   <th style="text-align:right;"> t </th>
   <th style="text-align:right;"> df_error </th>
   <th style="text-align:right;"> p </th>
   <th style="text-align:left;"> Effects </th>
   <th style="text-align:left;"> Group </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;background-color: white !important;"> (Intercept) </td>
   <td style="text-align:right;background-color: white !important;"> 0.600 </td>
   <td style="text-align:right;background-color: white !important;"> 0.060 </td>
   <td style="text-align:right;background-color: white !important;"> 0.95 </td>
   <td style="text-align:right;background-color: white !important;"> 0.482 </td>
   <td style="text-align:right;background-color: white !important;"> 0.718 </td>
   <td style="text-align:right;background-color: white !important;"> 10.065 </td>
   <td style="text-align:right;background-color: white !important;"> 184 </td>
   <td style="text-align:right;background-color: yellow !important;background-color: white !important;"> 0.000 </td>
   <td style="text-align:left;background-color: white !important;"> fixed </td>
   <td style="text-align:left;background-color: white !important;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> organism </td>
   <td style="text-align:right;"> -0.117 </td>
   <td style="text-align:right;"> 0.075 </td>
   <td style="text-align:right;"> 0.95 </td>
   <td style="text-align:right;"> -0.265 </td>
   <td style="text-align:right;"> 0.032 </td>
   <td style="text-align:right;"> -1.554 </td>
   <td style="text-align:right;"> 184 </td>
   <td style="text-align:right;background-color: yellow !important;"> 0.122 </td>
   <td style="text-align:left;"> fixed </td>
   <td style="text-align:left;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> SD (Intercept) </td>
   <td style="text-align:right;"> 0.000 </td>
   <td style="text-align:right;"> NA </td>
   <td style="text-align:right;"> 0.95 </td>
   <td style="text-align:right;"> NA </td>
   <td style="text-align:right;"> NA </td>
   <td style="text-align:right;"> NA </td>
   <td style="text-align:right;"> NA </td>
   <td style="text-align:right;background-color: yellow !important;"> NA </td>
   <td style="text-align:left;"> random </td>
   <td style="text-align:left;"> canteen </td>
  </tr>
  <tr>
   <td style="text-align:left;"> SD (Observations) </td>
   <td style="text-align:right;"> 0.499 </td>
   <td style="text-align:right;"> NA </td>
   <td style="text-align:right;"> 0.95 </td>
   <td style="text-align:right;"> NA </td>
   <td style="text-align:right;"> NA </td>
   <td style="text-align:right;"> NA </td>
   <td style="text-align:right;"> NA </td>
   <td style="text-align:right;background-color: yellow !important;"> NA </td>
   <td style="text-align:left;"> random </td>
   <td style="text-align:left;"> Residual </td>
  </tr>
</tbody>
</table>

---
.center[
![:scale 70%](media/which_test_flow.jpg)
]

---
.center[
![:scale 70%](media/which_test_flow_2.jpg)
]

---
# The only equation you will ever need
## The General Linear Model (GLM)

.ong_dk[
$$
`\begin{aligned}
\text{outcome}_i &= (\text{model}_i) + \text{error}_i \\
\text{outcome}_i &= \hat{b}_0 + \hat{b}_1\text{predictor}_{i} + \dots + \hat{b}_n\text{predictor}_{i} + \text{error}_i
\end{aligned}`
$$
]

`$\hat{b}_n$`

* Estimate of parameter for a predictor
	+ Direction/strength of relationship/effect
	+ Difference in means

`$\hat{b}_0$`

* Estimate of the value of the outcome when predictor(s) = 0 (intercept)

---
.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{volume}_{i} + e_i
\end{aligned}`
$$
]

---
class: center

.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{volume}_{i} + e_i
\end{aligned}`
$$
]

--
.pull-left[
![](linear_model_01_files/figure-html/unnamed-chunk-31-1.png)
]

---
class: center

.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{volume}_{i} + e_i
\end{aligned}`
$$
]

???

Let's zoom out of the plot a bit
---
class: center

.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{volume}_{i} + e_i
\end{aligned}`
$$
]

---
class: center

.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{volume}_{i} + e_i
\end{aligned}`
$$
]

---
class: center

.ong_dk[
$$
`\begin{aligned}
\hat{\text{ringing}}_i &= -37.12 + 10.45\text{volume}_{i}
\end{aligned}`
$$
]

???
Note that the intercept is a value that extrapolates well beyond the data.
We might want to take it with a pinch of salt.
The intercept is -37.12. That makes no sense - your ears can't ring for a minus amount of time. You'd expect the value to be zero in the population (0 db = 0 ringing). Illustrates that estimates are not accurate. Also, is it reasonable to assume a linear relationship here? Isn't it more likely that a range of low volumes ringing will be zero hours, and at a certain threshold ringing kicks in. It is almost certainly not linear along the whole range of volumes.

---
class: center

.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{musician}_{i} + e_i
\end{aligned}`
$$
]

![](linear_model_01_files/figure-html/unnamed-chunk-39-1.png)

---
class: center

.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{musician}_{i} + e_i
\end{aligned}`
$$
]

![](linear_model_01_files/figure-html/unnamed-chunk-40-1.png)

---
class: center

.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{musician}_{i} + e_i
\end{aligned}`
$$
]

![](linear_model_01_files/figure-html/unnamed-chunk-41-1.png)

---
class: middle, centre

![:scale 70%](media/ds5_fig_02.02_fitting_models.png)

$$
\text{outcome} = \text{model} + \text{error}_i
$$
</p>
]

$$
\text{outcome} = \text{estimated model} + \text{error}_i
$$
</p>
]

---
background-image: none
background-color: #000000

---
background-image: none
background-color: #000000

---
background-image: none
background-color: #000000

---
background-image: none
background-color: #000000

---

---
class: center, middle

![:scale 100%](media/dsr2_fig_06_04_lm_sample_01.png)
---
class: center, middle

![:scale 100%](media/dsr2_fig_06_04_lm_sample_02.png)

---
class: center, middle

![:scale 100%](media/dsr2_fig_06_04_lm_sample_03.png)

---
# Ordinary least squares (OLS) estimation

$$
\text{friends}_i = \hat{b}_0 + e_i
$$

]

.pull-left[
* With no predictors, we predict an outcome from only the intercept `$\hat{b}_0$`
* In this scenario `$\hat{b}_0$` will be the mean value of the outcome
* The model has one predicted value (the mean) for all observations
* Data: .blu[1, 3, 4, 3, 2]
]

---
## Guess 1: `$\hat{b}_0$` = 2

$$
`\begin{aligned}
\text{friends}_i &= \hat{b}_0 + \text{error}_i \\
\text{error}_i &= \text{friends}_i - \hat{b}_0
\end{aligned}`
$$

]

<div id="gcmiawpwnn" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>#gcmiawpwnn table {
  font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji';
  -webkit-font-smoothing: antialiased;
  -moz-osx-font-smoothing: grayscale;
}

#gcmiawpwnn thead, #gcmiawpwnn tbody, #gcmiawpwnn tfoot, #gcmiawpwnn tr, #gcmiawpwnn td, #gcmiawpwnn th {
  border-style: none;
}

#gcmiawpwnn p {
  margin: 0;
  padding: 0;
}

#gcmiawpwnn .gt_table {
  display: table;
  border-collapse: collapse;
  line-height: normal;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 16px;
  font-weight: normal;
  font-style: normal;
  background-color: #FFFFFF;
  width: auto;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #A8A8A8;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #A8A8A8;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
}

#gcmiawpwnn .gt_caption {
  padding-top: 4px;
  padding-bottom: 4px;
}

#gcmiawpwnn .gt_title {
  color: #333333;
  font-size: 125%;
  font-weight: initial;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#gcmiawpwnn .gt_subtitle {
  color: #333333;
  font-size: 85%;
  font-weight: initial;
  padding-top: 3px;
  padding-bottom: 5px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

#gcmiawpwnn .gt_heading {
  background-color: #FFFFFF;
  text-align: center;
  border-bottom-color: #FFFFFF;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#gcmiawpwnn .gt_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#gcmiawpwnn .gt_col_headings {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#gcmiawpwnn .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  padding-left: 5px;
  padding-right: 5px;
  overflow-x: hidden;
}

#gcmiawpwnn .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

#gcmiawpwnn .gt_column_spanner_outer:first-child {
  padding-left: 0;
}

#gcmiawpwnn .gt_column_spanner_outer:last-child {
  padding-right: 0;
}

#gcmiawpwnn .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 5px;
  overflow-x: hidden;
  display: inline-block;
  width: 100%;
}

#gcmiawpwnn .gt_spanner_row {
  border-bottom-style: hidden;
}

#gcmiawpwnn .gt_group_heading {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  text-align: left;
}

#gcmiawpwnn .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

#gcmiawpwnn .gt_from_md > :first-child {
  margin-top: 0;
}

#gcmiawpwnn .gt_from_md > :last-child {
  margin-bottom: 0;
}

#gcmiawpwnn .gt_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#gcmiawpwnn .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
}

#gcmiawpwnn .gt_stub_row_group {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
  vertical-align: top;
}

#gcmiawpwnn .gt_row_group_first td {
  border-top-width: 2px;
}

#gcmiawpwnn .gt_row_group_first th {
  border-top-width: 2px;
}

#gcmiawpwnn .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#gcmiawpwnn .gt_first_summary_row {
  border-top-style: solid;
  border-top-color: #D3D3D3;
}

#gcmiawpwnn .gt_first_summary_row.thick {
  border-top-width: 2px;
}

#gcmiawpwnn .gt_last_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#gcmiawpwnn .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#gcmiawpwnn .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#gcmiawpwnn .gt_last_grand_summary_row_top {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: double;
  border-bottom-width: 6px;
  border-bottom-color: #D3D3D3;
}

#gcmiawpwnn .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
}

#gcmiawpwnn .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#gcmiawpwnn .gt_footnotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#gcmiawpwnn .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#gcmiawpwnn .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#gcmiawpwnn .gt_sourcenote {
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#gcmiawpwnn .gt_left {
  text-align: left;
}

#gcmiawpwnn .gt_center {
  text-align: center;
}

#gcmiawpwnn .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#gcmiawpwnn .gt_font_normal {
  font-weight: normal;
}

#gcmiawpwnn .gt_font_bold {
  font-weight: bold;
}

#gcmiawpwnn .gt_font_italic {
  font-style: italic;
}

#gcmiawpwnn .gt_super {
  font-size: 65%;
}

#gcmiawpwnn .gt_footnote_marks {
  font-size: 75%;
  vertical-align: 0.4em;
  position: initial;
}

#gcmiawpwnn .gt_asterisk {
  font-size: 100%;
  vertical-align: 0;
}

#gcmiawpwnn .gt_indent_1 {
  text-indent: 5px;
}

#gcmiawpwnn .gt_indent_2 {
  text-indent: 10px;
}

#gcmiawpwnn .gt_indent_3 {
  text-indent: 15px;
}

#gcmiawpwnn .gt_indent_4 {
  text-indent: 20px;
}

#gcmiawpwnn .gt_indent_5 {
  text-indent: 25px;
}
</style>
<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Friends (Y)">Friends (Y)</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Estimate">Estimate</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Error">Error</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Squared error">Squared error</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="Friends (Y)" class="gt_row gt_center">1</td>
<td headers="Estimate" class="gt_row gt_center">2</td>
<td headers="Error" class="gt_row gt_center"></td>
<td headers="Squared error" class="gt_row gt_center"></td></tr>
    <tr><td headers="Friends (Y)" class="gt_row gt_center">2</td>
<td headers="Estimate" class="gt_row gt_center">2</td>
<td headers="Error" class="gt_row gt_center"></td>
<td headers="Squared error" class="gt_row gt_center"></td></tr>
    <tr><td headers="Friends (Y)" class="gt_row gt_center">3</td>
<td headers="Estimate" class="gt_row gt_center">2</td>
<td headers="Error" class="gt_row gt_center"></td>
<td headers="Squared error" class="gt_row gt_center"></td></tr>
    <tr><td headers="Friends (Y)" class="gt_row gt_center">3</td>
<td headers="Estimate" class="gt_row gt_center">2</td>
<td headers="Error" class="gt_row gt_center"></td>
<td headers="Squared error" class="gt_row gt_center"></td></tr>
    <tr><td headers="Friends (Y)" class="gt_row gt_center">4</td>
<td headers="Estimate" class="gt_row gt_center">2</td>
<td headers="Error" class="gt_row gt_center"></td>
<td headers="Squared error" class="gt_row gt_center"></td></tr>
  </tbody>
  
  
</table>
</div>

]

---
## Guess 1: `$\hat{b}_0$` = 2

$$
`\begin{aligned}
\text{friends}_i &= \hat{b}_0 + \text{error}_i \\
\text{error}_i &= \text{friends}_i - \hat{b}_0
\end{aligned}`
$$

]

<div id="oycjfyfowa" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>#oycjfyfowa table {
  font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji';
  -webkit-font-smoothing: antialiased;
  -moz-osx-font-smoothing: grayscale;
}

#oycjfyfowa thead, #oycjfyfowa tbody, #oycjfyfowa tfoot, #oycjfyfowa tr, #oycjfyfowa td, #oycjfyfowa th {
  border-style: none;
}

#oycjfyfowa p {
  margin: 0;
  padding: 0;
}

#oycjfyfowa .gt_table {
  display: table;
  border-collapse: collapse;
  line-height: normal;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 16px;
  font-weight: normal;
  font-style: normal;
  background-color: #FFFFFF;
  width: auto;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #A8A8A8;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #A8A8A8;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
}

#oycjfyfowa .gt_caption {
  padding-top: 4px;
  padding-bottom: 4px;
}

#oycjfyfowa .gt_title {
  color: #333333;
  font-size: 125%;
  font-weight: initial;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#oycjfyfowa .gt_subtitle {
  color: #333333;
  font-size: 85%;
  font-weight: initial;
  padding-top: 3px;
  padding-bottom: 5px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

#oycjfyfowa .gt_heading {
  background-color: #FFFFFF;
  text-align: center;
  border-bottom-color: #FFFFFF;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#oycjfyfowa .gt_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#oycjfyfowa .gt_col_headings {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#oycjfyfowa .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  padding-left: 5px;
  padding-right: 5px;
  overflow-x: hidden;
}

#oycjfyfowa .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

#oycjfyfowa .gt_column_spanner_outer:first-child {
  padding-left: 0;
}

#oycjfyfowa .gt_column_spanner_outer:last-child {
  padding-right: 0;
}

#oycjfyfowa .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 5px;
  overflow-x: hidden;
  display: inline-block;
  width: 100%;
}

#oycjfyfowa .gt_spanner_row {
  border-bottom-style: hidden;
}

#oycjfyfowa .gt_group_heading {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  text-align: left;
}

#oycjfyfowa .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

#oycjfyfowa .gt_from_md > :first-child {
  margin-top: 0;
}

#oycjfyfowa .gt_from_md > :last-child {
  margin-bottom: 0;
}

#oycjfyfowa .gt_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#oycjfyfowa .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
}

#oycjfyfowa .gt_stub_row_group {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
  vertical-align: top;
}

#oycjfyfowa .gt_row_group_first td {
  border-top-width: 2px;
}

#oycjfyfowa .gt_row_group_first th {
  border-top-width: 2px;
}

#oycjfyfowa .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#oycjfyfowa .gt_first_summary_row {
  border-top-style: solid;
  border-top-color: #D3D3D3;
}

#oycjfyfowa .gt_first_summary_row.thick {
  border-top-width: 2px;
}

#oycjfyfowa .gt_last_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#oycjfyfowa .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#oycjfyfowa .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#oycjfyfowa .gt_last_grand_summary_row_top {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: double;
  border-bottom-width: 6px;
  border-bottom-color: #D3D3D3;
}

#oycjfyfowa .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
}

#oycjfyfowa .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#oycjfyfowa .gt_footnotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#oycjfyfowa .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#oycjfyfowa .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#oycjfyfowa .gt_sourcenote {
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#oycjfyfowa .gt_left {
  text-align: left;
}

#oycjfyfowa .gt_center {
  text-align: center;
}

#oycjfyfowa .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#oycjfyfowa .gt_font_normal {
  font-weight: normal;
}

#oycjfyfowa .gt_font_bold {
  font-weight: bold;
}

#oycjfyfowa .gt_font_italic {
  font-style: italic;
}

#oycjfyfowa .gt_super {
  font-size: 65%;
}

#oycjfyfowa .gt_footnote_marks {
  font-size: 75%;
  vertical-align: 0.4em;
  position: initial;
}

#oycjfyfowa .gt_asterisk {
  font-size: 100%;
  vertical-align: 0;
}

#oycjfyfowa .gt_indent_1 {
  text-indent: 5px;
}

#oycjfyfowa .gt_indent_2 {
  text-indent: 10px;
}

#oycjfyfowa .gt_indent_3 {
  text-indent: 15px;
}

#oycjfyfowa .gt_indent_4 {
  text-indent: 20px;
}

#oycjfyfowa .gt_indent_5 {
  text-indent: 25px;
}
</style>
<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id=""></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="Friends (Y)">Friends (Y)</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="Estimate">Estimate</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="Error">Error</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="Squared error">Squared error</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><th id="stub_1_1" scope="row" class="gt_row gt_left gt_stub"></th>
<td headers="stub_1_1 Friends (Y)" class="gt_row gt_right">1</td>
<td headers="stub_1_1 Estimate" class="gt_row gt_right">2</td>
<td headers="stub_1_1 Error" class="gt_row gt_right">-1</td>
<td headers="stub_1_1 Squared error" class="gt_row gt_right">1</td></tr>
    <tr><th id="stub_1_2" scope="row" class="gt_row gt_left gt_stub"></th>
<td headers="stub_1_2 Friends (Y)" class="gt_row gt_right">2</td>
<td headers="stub_1_2 Estimate" class="gt_row gt_right">2</td>
<td headers="stub_1_2 Error" class="gt_row gt_right">0</td>
<td headers="stub_1_2 Squared error" class="gt_row gt_right">0</td></tr>
    <tr><th id="stub_1_3" scope="row" class="gt_row gt_left gt_stub"></th>
<td headers="stub_1_3 Friends (Y)" class="gt_row gt_right">3</td>
<td headers="stub_1_3 Estimate" class="gt_row gt_right">2</td>
<td headers="stub_1_3 Error" class="gt_row gt_right">1</td>
<td headers="stub_1_3 Squared error" class="gt_row gt_right">1</td></tr>
    <tr><th id="stub_1_4" scope="row" class="gt_row gt_left gt_stub"></th>
<td headers="stub_1_4 Friends (Y)" class="gt_row gt_right">3</td>
<td headers="stub_1_4 Estimate" class="gt_row gt_right">2</td>
<td headers="stub_1_4 Error" class="gt_row gt_right">1</td>
<td headers="stub_1_4 Squared error" class="gt_row gt_right">1</td></tr>
    <tr><th id="stub_1_5" scope="row" class="gt_row gt_left gt_stub"></th>
<td headers="stub_1_5 Friends (Y)" class="gt_row gt_right">4</td>
<td headers="stub_1_5 Estimate" class="gt_row gt_right">2</td>
<td headers="stub_1_5 Error" class="gt_row gt_right">2</td>
<td headers="stub_1_5 Squared error" class="gt_row gt_right">4</td></tr>
    <tr><th id="grand_summary_stub_1" scope="row" class="gt_row gt_left gt_stub gt_grand_summary_row gt_first_grand_summary_row gt_last_summary_row">Total</th>
<td headers="grand_summary_stub_1 Friends (Y)" class="gt_row gt_right gt_grand_summary_row gt_first_grand_summary_row gt_last_summary_row">—</td>
<td headers="grand_summary_stub_1 Estimate" class="gt_row gt_right gt_grand_summary_row gt_first_grand_summary_row gt_last_summary_row">—</td>
<td headers="grand_summary_stub_1 Error" class="gt_row gt_right gt_grand_summary_row gt_first_grand_summary_row gt_last_summary_row">—</td>
<td headers="grand_summary_stub_1 Squared error" class="gt_row gt_right gt_grand_summary_row gt_first_grand_summary_row gt_last_summary_row">7</td></tr>
  </tbody>
  
  
</table>
</div>

]

---
## Guess 2: `$\hat{b}_0$` = 4

$$
`\begin{aligned}
\text{friends}_i &= \hat{b}_0 + \text{error}_i \\
\text{error}_i &= \text{friends}_i - \hat{b}_0
\end{aligned}`
$$

]

<div id="rbghmilijz" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>#rbghmilijz table {
  font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji';
  -webkit-font-smoothing: antialiased;
  -moz-osx-font-smoothing: grayscale;
}

#rbghmilijz thead, #rbghmilijz tbody, #rbghmilijz tfoot, #rbghmilijz tr, #rbghmilijz td, #rbghmilijz th {
  border-style: none;
}

#rbghmilijz p {
  margin: 0;
  padding: 0;
}

#rbghmilijz .gt_table {
  display: table;
  border-collapse: collapse;
  line-height: normal;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 16px;
  font-weight: normal;
  font-style: normal;
  background-color: #FFFFFF;
  width: auto;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #A8A8A8;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #A8A8A8;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
}

#rbghmilijz .gt_caption {
  padding-top: 4px;
  padding-bottom: 4px;
}

#rbghmilijz .gt_title {
  color: #333333;
  font-size: 125%;
  font-weight: initial;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#rbghmilijz .gt_subtitle {
  color: #333333;
  font-size: 85%;
  font-weight: initial;
  padding-top: 3px;
  padding-bottom: 5px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

#rbghmilijz .gt_heading {
  background-color: #FFFFFF;
  text-align: center;
  border-bottom-color: #FFFFFF;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#rbghmilijz .gt_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#rbghmilijz .gt_col_headings {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#rbghmilijz .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  padding-left: 5px;
  padding-right: 5px;
  overflow-x: hidden;
}

#rbghmilijz .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

#rbghmilijz .gt_column_spanner_outer:first-child {
  padding-left: 0;
}

#rbghmilijz .gt_column_spanner_outer:last-child {
  padding-right: 0;
}

#rbghmilijz .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 5px;
  overflow-x: hidden;
  display: inline-block;
  width: 100%;
}

#rbghmilijz .gt_spanner_row {
  border-bottom-style: hidden;
}

#rbghmilijz .gt_group_heading {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  text-align: left;
}

#rbghmilijz .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

#rbghmilijz .gt_from_md > :first-child {
  margin-top: 0;
}

#rbghmilijz .gt_from_md > :last-child {
  margin-bottom: 0;
}

#rbghmilijz .gt_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#rbghmilijz .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
}

#rbghmilijz .gt_stub_row_group {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
  vertical-align: top;
}

#rbghmilijz .gt_row_group_first td {
  border-top-width: 2px;
}

#rbghmilijz .gt_row_group_first th {
  border-top-width: 2px;
}

#rbghmilijz .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#rbghmilijz .gt_first_summary_row {
  border-top-style: solid;
  border-top-color: #D3D3D3;
}

#rbghmilijz .gt_first_summary_row.thick {
  border-top-width: 2px;
}

#rbghmilijz .gt_last_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#rbghmilijz .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#rbghmilijz .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#rbghmilijz .gt_last_grand_summary_row_top {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: double;
  border-bottom-width: 6px;
  border-bottom-color: #D3D3D3;
}

#rbghmilijz .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
}

#rbghmilijz .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#rbghmilijz .gt_footnotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#rbghmilijz .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#rbghmilijz .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#rbghmilijz .gt_sourcenote {
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#rbghmilijz .gt_left {
  text-align: left;
}

#rbghmilijz .gt_center {
  text-align: center;
}

#rbghmilijz .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#rbghmilijz .gt_font_normal {
  font-weight: normal;
}

#rbghmilijz .gt_font_bold {
  font-weight: bold;
}

#rbghmilijz .gt_font_italic {
  font-style: italic;
}

#rbghmilijz .gt_super {
  font-size: 65%;
}

#rbghmilijz .gt_footnote_marks {
  font-size: 75%;
  vertical-align: 0.4em;
  position: initial;
}

#rbghmilijz .gt_asterisk {
  font-size: 100%;
  vertical-align: 0;
}

#rbghmilijz .gt_indent_1 {
  text-indent: 5px;
}

#rbghmilijz .gt_indent_2 {
  text-indent: 10px;
}

#rbghmilijz .gt_indent_3 {
  text-indent: 15px;
}

#rbghmilijz .gt_indent_4 {
  text-indent: 20px;
}

#rbghmilijz .gt_indent_5 {
  text-indent: 25px;
}
</style>
<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="Friends (Y)">Friends (Y)</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="Estimate">Estimate</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="Error">Error</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="Squared error">Squared error</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="Friends (Y)" class="gt_row gt_right">1</td>
<td headers="Estimate" class="gt_row gt_right">4</td>
<td headers="Error" class="gt_row gt_right"></td>
<td headers="Squared error" class="gt_row gt_right"></td></tr>
    <tr><td headers="Friends (Y)" class="gt_row gt_right">2</td>
<td headers="Estimate" class="gt_row gt_right">4</td>
<td headers="Error" class="gt_row gt_right"></td>
<td headers="Squared error" class="gt_row gt_right"></td></tr>
    <tr><td headers="Friends (Y)" class="gt_row gt_right">3</td>
<td headers="Estimate" class="gt_row gt_right">4</td>
<td headers="Error" class="gt_row gt_right"></td>
<td headers="Squared error" class="gt_row gt_right"></td></tr>
    <tr><td headers="Friends (Y)" class="gt_row gt_right">3</td>
<td headers="Estimate" class="gt_row gt_right">4</td>
<td headers="Error" class="gt_row gt_right"></td>
<td headers="Squared error" class="gt_row gt_right"></td></tr>
    <tr><td headers="Friends (Y)" class="gt_row gt_right">4</td>
<td headers="Estimate" class="gt_row gt_right">4</td>
<td headers="Error" class="gt_row gt_right"></td>
<td headers="Squared error" class="gt_row gt_right"></td></tr>
  </tbody>
  
  
</table>
</div>

]

---
## Guess 2: `$\hat{b}_0$` = 4

$$
`\begin{aligned}
\text{friends}_i &= \hat{b}_0 + \text{error}_i \\
\text{error}_i &= \text{friends}_i - \hat{b}_0
\end{aligned}`
$$

]

<div id="tmfyqddasf" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>#tmfyqddasf table {
  font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji';
  -webkit-font-smoothing: antialiased;
  -moz-osx-font-smoothing: grayscale;
}

#tmfyqddasf thead, #tmfyqddasf tbody, #tmfyqddasf tfoot, #tmfyqddasf tr, #tmfyqddasf td, #tmfyqddasf th {
  border-style: none;
}

#tmfyqddasf p {
  margin: 0;
  padding: 0;
}

#tmfyqddasf .gt_table {
  display: table;
  border-collapse: collapse;
  line-height: normal;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 16px;
  font-weight: normal;
  font-style: normal;
  background-color: #FFFFFF;
  width: auto;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #A8A8A8;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #A8A8A8;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
}

#tmfyqddasf .gt_caption {
  padding-top: 4px;
  padding-bottom: 4px;
}

#tmfyqddasf .gt_title {
  color: #333333;
  font-size: 125%;
  font-weight: initial;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#tmfyqddasf .gt_subtitle {
  color: #333333;
  font-size: 85%;
  font-weight: initial;
  padding-top: 3px;
  padding-bottom: 5px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

#tmfyqddasf .gt_heading {
  background-color: #FFFFFF;
  text-align: center;
  border-bottom-color: #FFFFFF;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#tmfyqddasf .gt_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#tmfyqddasf .gt_col_headings {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#tmfyqddasf .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  padding-left: 5px;
  padding-right: 5px;
  overflow-x: hidden;
}

#tmfyqddasf .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

#tmfyqddasf .gt_column_spanner_outer:first-child {
  padding-left: 0;
}

#tmfyqddasf .gt_column_spanner_outer:last-child {
  padding-right: 0;
}

#tmfyqddasf .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 5px;
  overflow-x: hidden;
  display: inline-block;
  width: 100%;
}

#tmfyqddasf .gt_spanner_row {
  border-bottom-style: hidden;
}

#tmfyqddasf .gt_group_heading {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  text-align: left;
}

#tmfyqddasf .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

#tmfyqddasf .gt_from_md > :first-child {
  margin-top: 0;
}

#tmfyqddasf .gt_from_md > :last-child {
  margin-bottom: 0;
}

#tmfyqddasf .gt_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#tmfyqddasf .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
}

#tmfyqddasf .gt_stub_row_group {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
  vertical-align: top;
}

#tmfyqddasf .gt_row_group_first td {
  border-top-width: 2px;
}

#tmfyqddasf .gt_row_group_first th {
  border-top-width: 2px;
}

#tmfyqddasf .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#tmfyqddasf .gt_first_summary_row {
  border-top-style: solid;
  border-top-color: #D3D3D3;
}

#tmfyqddasf .gt_first_summary_row.thick {
  border-top-width: 2px;
}

#tmfyqddasf .gt_last_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#tmfyqddasf .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#tmfyqddasf .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#tmfyqddasf .gt_last_grand_summary_row_top {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: double;
  border-bottom-width: 6px;
  border-bottom-color: #D3D3D3;
}

#tmfyqddasf .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
}

#tmfyqddasf .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#tmfyqddasf .gt_footnotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#tmfyqddasf .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#tmfyqddasf .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#tmfyqddasf .gt_sourcenote {
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#tmfyqddasf .gt_left {
  text-align: left;
}

#tmfyqddasf .gt_center {
  text-align: center;
}

#tmfyqddasf .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#tmfyqddasf .gt_font_normal {
  font-weight: normal;
}

#tmfyqddasf .gt_font_bold {
  font-weight: bold;
}

#tmfyqddasf .gt_font_italic {
  font-style: italic;
}

#tmfyqddasf .gt_super {
  font-size: 65%;
}

#tmfyqddasf .gt_footnote_marks {
  font-size: 75%;
  vertical-align: 0.4em;
  position: initial;
}

#tmfyqddasf .gt_asterisk {
  font-size: 100%;
  vertical-align: 0;
}

#tmfyqddasf .gt_indent_1 {
  text-indent: 5px;
}

#tmfyqddasf .gt_indent_2 {
  text-indent: 10px;
}

#tmfyqddasf .gt_indent_3 {
  text-indent: 15px;
}

#tmfyqddasf .gt_indent_4 {
  text-indent: 20px;
}

#tmfyqddasf .gt_indent_5 {
  text-indent: 25px;
}
</style>
<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id=""></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Friends (Y)">Friends (Y)</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Estimate">Estimate</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Error">Error</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Squared error">Squared error</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><th id="stub_1_1" scope="row" class="gt_row gt_center gt_stub"></th>
<td headers="stub_1_1 Friends (Y)" class="gt_row gt_center">1</td>
<td headers="stub_1_1 Estimate" class="gt_row gt_center">4</td>
<td headers="stub_1_1 Error" class="gt_row gt_center">-3</td>
<td headers="stub_1_1 Squared error" class="gt_row gt_center">9</td></tr>
    <tr><th id="stub_1_2" scope="row" class="gt_row gt_center gt_stub"></th>
<td headers="stub_1_2 Friends (Y)" class="gt_row gt_center">2</td>
<td headers="stub_1_2 Estimate" class="gt_row gt_center">4</td>
<td headers="stub_1_2 Error" class="gt_row gt_center">-2</td>
<td headers="stub_1_2 Squared error" class="gt_row gt_center">4</td></tr>
    <tr><th id="stub_1_3" scope="row" class="gt_row gt_center gt_stub"></th>
<td headers="stub_1_3 Friends (Y)" class="gt_row gt_center">3</td>
<td headers="stub_1_3 Estimate" class="gt_row gt_center">4</td>
<td headers="stub_1_3 Error" class="gt_row gt_center">-1</td>
<td headers="stub_1_3 Squared error" class="gt_row gt_center">1</td></tr>
    <tr><th id="stub_1_4" scope="row" class="gt_row gt_center gt_stub"></th>
<td headers="stub_1_4 Friends (Y)" class="gt_row gt_center">3</td>
<td headers="stub_1_4 Estimate" class="gt_row gt_center">4</td>
<td headers="stub_1_4 Error" class="gt_row gt_center">-1</td>
<td headers="stub_1_4 Squared error" class="gt_row gt_center">1</td></tr>
    <tr><th id="stub_1_5" scope="row" class="gt_row gt_center gt_stub"></th>
<td headers="stub_1_5 Friends (Y)" class="gt_row gt_center">4</td>
<td headers="stub_1_5 Estimate" class="gt_row gt_center">4</td>
<td headers="stub_1_5 Error" class="gt_row gt_center">0</td>
<td headers="stub_1_5 Squared error" class="gt_row gt_center">0</td></tr>
    <tr><th id="grand_summary_stub_1" scope="row" class="gt_row gt_left gt_stub gt_grand_summary_row gt_first_grand_summary_row gt_last_summary_row">Total</th>
<td headers="grand_summary_stub_1 Friends (Y)" class="gt_row gt_center gt_grand_summary_row gt_first_grand_summary_row gt_last_summary_row">—</td>
<td headers="grand_summary_stub_1 Estimate" class="gt_row gt_center gt_grand_summary_row gt_first_grand_summary_row gt_last_summary_row">—</td>
<td headers="grand_summary_stub_1 Error" class="gt_row gt_center gt_grand_summary_row gt_first_grand_summary_row gt_last_summary_row">—</td>
<td headers="grand_summary_stub_1 Squared error" class="gt_row gt_center gt_grand_summary_row gt_first_grand_summary_row gt_last_summary_row">15</td></tr>
  </tbody>
  
  
</table>
</div>

]

---
class: center

![:scale 60%](media/ols_animate.gif)
---
class: center

![](linear_model_01_files/figure-html/unnamed-chunk-56-1.png)

---
class: center

![](linear_model_01_files/figure-html/unnamed-chunk-57-1.png)

---
class: center

![](linear_model_01_files/figure-html/unnamed-chunk-58-1.png)

---
## OLS estimate: `$\hat{b}_0$` = 2.6

$$
`\begin{aligned}
\text{friends}_i &= \hat{b}_0 + \text{error}_i \\
\text{error}_i &= \text{friends}_i - \hat{b}_0
\end{aligned}`
$$

]

.pull-left[
<div id="wuaftgigxd" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>#wuaftgigxd table {
  font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji';
  -webkit-font-smoothing: antialiased;
  -moz-osx-font-smoothing: grayscale;
}

#wuaftgigxd thead, #wuaftgigxd tbody, #wuaftgigxd tfoot, #wuaftgigxd tr, #wuaftgigxd td, #wuaftgigxd th {
  border-style: none;
}

#wuaftgigxd p {
  margin: 0;
  padding: 0;
}

#wuaftgigxd .gt_table {
  display: table;
  border-collapse: collapse;
  line-height: normal;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 16px;
  font-weight: normal;
  font-style: normal;
  background-color: #FFFFFF;
  width: auto;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #A8A8A8;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #A8A8A8;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
}

#wuaftgigxd .gt_caption {
  padding-top: 4px;
  padding-bottom: 4px;
}

#wuaftgigxd .gt_title {
  color: #333333;
  font-size: 125%;
  font-weight: initial;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#wuaftgigxd .gt_subtitle {
  color: #333333;
  font-size: 85%;
  font-weight: initial;
  padding-top: 3px;
  padding-bottom: 5px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

#wuaftgigxd .gt_heading {
  background-color: #FFFFFF;
  text-align: center;
  border-bottom-color: #FFFFFF;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#wuaftgigxd .gt_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#wuaftgigxd .gt_col_headings {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#wuaftgigxd .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  padding-left: 5px;
  padding-right: 5px;
  overflow-x: hidden;
}

#wuaftgigxd .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

#wuaftgigxd .gt_column_spanner_outer:first-child {
  padding-left: 0;
}

#wuaftgigxd .gt_column_spanner_outer:last-child {
  padding-right: 0;
}

#wuaftgigxd .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 5px;
  overflow-x: hidden;
  display: inline-block;
  width: 100%;
}

#wuaftgigxd .gt_spanner_row {
  border-bottom-style: hidden;
}

#wuaftgigxd .gt_group_heading {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  text-align: left;
}

#wuaftgigxd .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

#wuaftgigxd .gt_from_md > :first-child {
  margin-top: 0;
}

#wuaftgigxd .gt_from_md > :last-child {
  margin-bottom: 0;
}

#wuaftgigxd .gt_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#wuaftgigxd .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
}

#wuaftgigxd .gt_stub_row_group {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
  vertical-align: top;
}

#wuaftgigxd .gt_row_group_first td {
  border-top-width: 2px;
}

#wuaftgigxd .gt_row_group_first th {
  border-top-width: 2px;
}

#wuaftgigxd .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#wuaftgigxd .gt_first_summary_row {
  border-top-style: solid;
  border-top-color: #D3D3D3;
}

#wuaftgigxd .gt_first_summary_row.thick {
  border-top-width: 2px;
}

#wuaftgigxd .gt_last_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#wuaftgigxd .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#wuaftgigxd .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#wuaftgigxd .gt_last_grand_summary_row_top {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: double;
  border-bottom-width: 6px;
  border-bottom-color: #D3D3D3;
}

#wuaftgigxd .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
}

#wuaftgigxd .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#wuaftgigxd .gt_footnotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#wuaftgigxd .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#wuaftgigxd .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#wuaftgigxd .gt_sourcenote {
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#wuaftgigxd .gt_left {
  text-align: left;
}

#wuaftgigxd .gt_center {
  text-align: center;
}

#wuaftgigxd .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#wuaftgigxd .gt_font_normal {
  font-weight: normal;
}

#wuaftgigxd .gt_font_bold {
  font-weight: bold;
}

#wuaftgigxd .gt_font_italic {
  font-style: italic;
}

#wuaftgigxd .gt_super {
  font-size: 65%;
}

#wuaftgigxd .gt_footnote_marks {
  font-size: 75%;
  vertical-align: 0.4em;
  position: initial;
}

#wuaftgigxd .gt_asterisk {
  font-size: 100%;
  vertical-align: 0;
}

#wuaftgigxd .gt_indent_1 {
  text-indent: 5px;
}

#wuaftgigxd .gt_indent_2 {
  text-indent: 10px;
}

#wuaftgigxd .gt_indent_3 {
  text-indent: 15px;
}

#wuaftgigxd .gt_indent_4 {
  text-indent: 20px;
}

#wuaftgigxd .gt_indent_5 {
  text-indent: 25px;
}
</style>
<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id=""></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Friends (Y)">Friends (Y)</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Estimate">Estimate</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Error">Error</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Squared error">Squared error</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><th id="stub_1_1" scope="row" class="gt_row gt_left gt_stub"></th>
<td headers="stub_1_1 Friends (Y)" class="gt_row gt_center">1</td>
<td headers="stub_1_1 Estimate" class="gt_row gt_center">2.6</td>
<td headers="stub_1_1 Error" class="gt_row gt_center">-1.6</td>
<td headers="stub_1_1 Squared error" class="gt_row gt_center">2.56</td></tr>
    <tr><th id="stub_1_2" scope="row" class="gt_row gt_left gt_stub"></th>
<td headers="stub_1_2 Friends (Y)" class="gt_row gt_center">2</td>
<td headers="stub_1_2 Estimate" class="gt_row gt_center">2.6</td>
<td headers="stub_1_2 Error" class="gt_row gt_center">-0.6</td>
<td headers="stub_1_2 Squared error" class="gt_row gt_center">0.36</td></tr>
    <tr><th id="stub_1_3" scope="row" class="gt_row gt_left gt_stub"></th>
<td headers="stub_1_3 Friends (Y)" class="gt_row gt_center">3</td>
<td headers="stub_1_3 Estimate" class="gt_row gt_center">2.6</td>
<td headers="stub_1_3 Error" class="gt_row gt_center">0.4</td>
<td headers="stub_1_3 Squared error" class="gt_row gt_center">0.16</td></tr>
    <tr><th id="stub_1_4" scope="row" class="gt_row gt_left gt_stub"></th>
<td headers="stub_1_4 Friends (Y)" class="gt_row gt_center">3</td>
<td headers="stub_1_4 Estimate" class="gt_row gt_center">2.6</td>
<td headers="stub_1_4 Error" class="gt_row gt_center">0.4</td>
<td headers="stub_1_4 Squared error" class="gt_row gt_center">0.16</td></tr>
    <tr><th id="stub_1_5" scope="row" class="gt_row gt_left gt_stub"></th>
<td headers="stub_1_5 Friends (Y)" class="gt_row gt_center">4</td>
<td headers="stub_1_5 Estimate" class="gt_row gt_center">2.6</td>
<td headers="stub_1_5 Error" class="gt_row gt_center">1.4</td>
<td headers="stub_1_5 Squared error" class="gt_row gt_center">1.96</td></tr>
    <tr><th id="grand_summary_stub_1" scope="row" class="gt_row gt_left gt_stub gt_grand_summary_row gt_first_grand_summary_row gt_last_summary_row">Total</th>
<td headers="grand_summary_stub_1 Friends (Y)" class="gt_row gt_center gt_grand_summary_row gt_first_grand_summary_row gt_last_summary_row">—</td>
<td headers="grand_summary_stub_1 Estimate" class="gt_row gt_center gt_grand_summary_row gt_first_grand_summary_row gt_last_summary_row">—</td>
<td headers="grand_summary_stub_1 Error" class="gt_row gt_center gt_grand_summary_row gt_first_grand_summary_row gt_last_summary_row">—</td>
<td headers="grand_summary_stub_1 Squared error" class="gt_row gt_center gt_grand_summary_row gt_first_grand_summary_row gt_last_summary_row">5.2</td></tr>
  </tbody>
  
  
</table>
</div>

]

---

* Data: 1, 3, 4, 3, 2

* Add up the scores

* Divide by the number of scores, *n*
.ong_dk[
$$
`\begin{aligned}
\frac{\sum\limits^n_{i = 1}x_i}{n} &= \frac{13}{5} \\
&= 2.6
\end{aligned}`
$$
]

---
# Summary

* People try to make statistics seem complicated but boils down to simple ideas:
    - We predict one variable from a model containing one or more predictors
    - There is always error in prediction
    - The model you fit represents hypotheses
    - The model you fit is typically a variation on the the linear model (GLM)
    - If you understand this one model, you understand most of psychological statistics

--
* Parameter estimates (*b*)
    - Tell us about the shape of the model
    - Tell us about size and direction of relationship between predictor(s) and outcome
    - They are estimated from the sample data

--
* Estimation
    - Ordinary least squares (OLS) estimation is one (of many) methods for estimating parameters
    - The mean is an example of an OLS estimator