ECON 3818

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# ECON 3818
## Chapter 23
### Kyle Butts
### 26 September 2021

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## Chapter 23: Comparing Two Proportions

---
# Notation
We will use notation similar to that used in our study of two-sample t-statistics.

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#wynycqagdq .gt_sourcenote {
  font-size: 12px;
  padding: 4px;
}

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

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

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

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

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

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

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

#wynycqagdq .gt_footnote_marks {
  font-style: italic;
  font-weight: normal;
  font-size: 65%;
}
</style>
<table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">Population</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">Pop. Proportion</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">Sample Size</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">Sample Proportion</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_right">1</td>
<td class="gt_row gt_left">$ p_1 $</td>
<td class="gt_row gt_left">$ n_1 $</td>
<td class="gt_row gt_left">$ \hat{p}_1 $</td></tr>
    <tr><td class="gt_row gt_right">2</td>
<td class="gt_row gt_left">$ p_2 $</td>
<td class="gt_row gt_left">$ n_2 $</td>
<td class="gt_row gt_left">$ \hat{p}_2 $</td></tr>
  </tbody>
  
  
</table>
</div>

---
# Sampling Distribution of Sample Proportion Review
`$X \sim B(1, p)$` is the underlying variable.

$$ \hat{p} = \frac{\sum \text{\# of successes}}{n} $$

The sample distribution of `$\hat{p}$` with population proportion `$p_0$`:

$$ \hat{p} \sim N( p_0, \frac{p_0(1-p_0)}{n}) $$

---
# Sampling Distribution of a Difference between Proportions

To use `$\hat{p}_1 - \hat{p}_2$` for inference we use the following information:

- When the samples are large, the distribution of `$\hat{p}_1 - \hat{p}_2$` is .hi.purple[approximately normal]

- The .hi.purple[mean] of the sampling distribution is: `$p_1 - p_2$`

- Assuming the two populations are independent, the .hi.purple[standard deviation] of the distribution is: 
$$ 
\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}} 
$$

---
# Normal Distribution Review
If `$X_1 \sim N(\mu_1, \sigma_1^2)$` and `$X_2 \sim N(\mu_2, \sigma_2^2)$` are normally distributed and indepdent, then `$X_1 - X_2$` is normally distributed,

$$ E(X_1 - X_2) = \mu_1 - \mu_2, $$

$$ Var(X_1 - X_2) = \sigma_1^2 + \sigma_2^2 $$

---
# Large-Sample Confidence Intervals for Comparing Proportions
Using the equation for standard error:
$$ 
SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}} 
$$ 
The confidence interval is constructed as:
$$ 
\hat{p}_1 - \hat{p}_2 \pm Z^* SE,
$$

where `$Z^*$` is the associated critical value.

---
# Example
Construct a 95\% confidence interval for the following difference in proportions:

<div id="cxqpvpcavs" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>@import url("https://fonts.googleapis.com/css2?family=Fira+Code:ital,wght@0,100;0,200;0,300;0,400;0,500;0,600;0,700;0,800;0,900;1,100;1,200;1,300;1,400;1,500;1,600;1,700;1,800;1,900&display=swap");
html {
  font-family: 'Fira Code', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen, Ubuntu, Cantarell, 'Helvetica Neue', 'Fira Sans', 'Droid Sans', Arial, sans-serif;
}

#cxqpvpcavs .gt_table {
  display: table;
  border-collapse: collapse;
  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: 3px;
  border-top-color: transparent;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: transparent;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
}

#cxqpvpcavs .gt_heading {
  background-color: #FFFFFF;
  text-align: left;
  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;
}

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

#cxqpvpcavs .gt_subtitle {
  color: #333333;
  font-size: 85%;
  font-weight: initial;
  padding-top: 0;
  padding-bottom: 4px;
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

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

#cxqpvpcavs .gt_col_headings {
  border-top-style: solid;
  border-top-width: 3px;
  border-top-color: transparent;
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: black;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#cxqpvpcavs .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  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;
}

#cxqpvpcavs .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

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

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

#cxqpvpcavs .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: black;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  overflow-x: hidden;
  display: inline-block;
  width: 100%;
}

#cxqpvpcavs .gt_group_heading {
  padding: 8px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  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;
}

#cxqpvpcavs .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  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;
}

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

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

#cxqpvpcavs .gt_row {
  padding-top: 5px;
  padding-bottom: 5px;
  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;
}

#cxqpvpcavs .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 12px;
}

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

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

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

#cxqpvpcavs .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;
}

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

#cxqpvpcavs .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;
}

#cxqpvpcavs .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;
}

#cxqpvpcavs .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding: 4px;
}

#cxqpvpcavs .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;
}

#cxqpvpcavs .gt_sourcenote {
  font-size: 12px;
  padding: 4px;
}

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

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

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

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

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

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

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

#cxqpvpcavs .gt_footnote_marks {
  font-style: italic;
  font-weight: normal;
  font-size: 65%;
}
</style>
<table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Population</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">No. Successes</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Sample Size</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Sample Proportion</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_center">1</td>
<td class="gt_row gt_center">75</td>
<td class="gt_row gt_center">100</td>
<td class="gt_row gt_center">$ \hat{p}_1 $ = 0.75</td></tr>
    <tr><td class="gt_row gt_center">2</td>
<td class="gt_row gt_center">56</td>
<td class="gt_row gt_center">100</td>
<td class="gt_row gt_center">$ \hat{p}_2 $ = 0.56</td></tr>
  </tbody>
  
  
</table>
</div>

$$ 
SE=\sqrt{\frac{(0.75)(0.25)}{100}+\frac{(0.56)(0.44)}{100}}=0.0659
$$

Confidence interval = `$(0.75-0.56) \pm (1.96)(0.0659) \implies [0.06, 0.32]$`

---
# Significance Tests for Comparing Proportions
$$ 
H_0: p_1 - p_2 = 0 
$$

$$
H_1: p_1 - p_2 \neq 0 
$$

In order to test the hypothesis, we must first calculated the .hi.purple[pooled sample proportion]

$$ 
\hat{p}=\frac{\text{number of successes in both samples combined}}{\text{number of individuals in both samples combined}} 
$$

Then we use the following z-statistic:

$$ 
\frac{\hat{p}_1-\hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}} 
$$

---
# Example

<div id="cltugtopeu" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>@import url("https://fonts.googleapis.com/css2?family=Fira+Code:ital,wght@0,100;0,200;0,300;0,400;0,500;0,600;0,700;0,800;0,900;1,100;1,200;1,300;1,400;1,500;1,600;1,700;1,800;1,900&display=swap");
html {
  font-family: 'Fira Code', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen, Ubuntu, Cantarell, 'Helvetica Neue', 'Fira Sans', 'Droid Sans', Arial, sans-serif;
}

#cltugtopeu .gt_table {
  display: table;
  border-collapse: collapse;
  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: 3px;
  border-top-color: transparent;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: transparent;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
}

#cltugtopeu .gt_heading {
  background-color: #FFFFFF;
  text-align: left;
  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;
}

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

#cltugtopeu .gt_subtitle {
  color: #333333;
  font-size: 85%;
  font-weight: initial;
  padding-top: 0;
  padding-bottom: 4px;
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

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

#cltugtopeu .gt_col_headings {
  border-top-style: solid;
  border-top-width: 3px;
  border-top-color: transparent;
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: black;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#cltugtopeu .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  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;
}

#cltugtopeu .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

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

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

#cltugtopeu .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: black;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  overflow-x: hidden;
  display: inline-block;
  width: 100%;
}

#cltugtopeu .gt_group_heading {
  padding: 8px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  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;
}

#cltugtopeu .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  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;
}

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

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

#cltugtopeu .gt_row {
  padding-top: 5px;
  padding-bottom: 5px;
  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;
}

#cltugtopeu .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 12px;
}

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

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

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

#cltugtopeu .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;
}

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

#cltugtopeu .gt_table_body {
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<table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Population</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">No. Successes</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Sample Size</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Sample Proportion</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_center">1</td>
<td class="gt_row gt_center">212</td>
<td class="gt_row gt_center">616</td>
<td class="gt_row gt_center">$ \hat{p}_1 $ = 0.344</td></tr>
    <tr><td class="gt_row gt_center">2</td>
<td class="gt_row gt_center">7</td>
<td class="gt_row gt_center">49</td>
<td class="gt_row gt_center">$ \hat{p}_2 $ = 0.143</td></tr>
  </tbody>
  
  
</table>
</div>

- Calculate `$\hat{p}$`

$$ 
\hat{p} = \frac{212+7}{616+49} = 0.329 
$$
- Calculate `$Z$`-statistic

--
$$ 
Z= \frac{0.344-0.143}{\sqrt{(0.329)(0.671) \left(\frac{1}{616}+\frac{1}{49}\right)} } = 2.88
$$

---
# Example
.subheader.alice[Continued]

The z-statistic was 2.88, and we have a two-tailed alternative hypothesis. Therefore:

$$ 
\text{p-value } = 2\cdot P(Z>2.88) = 2\cdot 0.002 = 0.004 
$$

Therefore we reject null at `$\alpha = 0.05$`, since p-value `$< \alpha$`