BANL 6100: Business Analytics

class: center, middle, inverse, title-slide

.title[
# BANL 6100: Business Analytics
]
.subtitle[
## Inference about a Population Proportion
]
.author[
### Mehmet Balcilar </br> <a href="mailto:mbalcilar@newhaven.edu" class="email">mbalcilar@newhaven.edu</a>
]
.institute[
### Univeristy of New Haven
]
.date[
### 2023-09-28 (updated: 2024-01-28)
]

---

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---
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# Inference about a Population Proportion

---
## Inference about a Population Proportion

Previously we have discussed making inference in population *means*

This chapter talks about questions where we're interested in the proportion of an outcome

.bf.alice[Single population proportion]

- .ex[Examples:] Proportion of people voting for a candidate; percent of people who are vaccinated; percent of people who support an issue; etc.

.bf.alice[Comparing two population proportions]

- .ex[Examples:] Is there a difference between the proportion of male students and proportion of female students that smoke cigarettes; Do Republicans and Democrats differ in their support for policy X; etc.

---
## The Sample Proportion, `$\hat{p}$`

The statistic that estimates the population proportion, `$p$`, is the .hi.coral[sample proportion]:

$$
\coral{\hat{p}} = \frac{\text{number of “successes" in the sample}}{n}
$$

For example:

Say we want to estimate the proportion of adults who smokes. To estimate this proportion, a researcher collected survey data and contacted 2673 people, and 170 said they smoke:

$$
\coral{\hat{p}} = \frac{170}{2673} = 0.0636
$$

---
## Sampling Distribution of a Sample Proportion

.subheader.alice[Binomial Distribution Review]

We can think of binary random variables (take only two values) as a Bernoulli distribution:

- Assign one outcome 0 and the other outcome 1

- `$X \sim B(1,p)$`

- This means `$p$` is the **unobserved** probability of outcome 1 occuring

We use the sample statistic:
`$$\coral{\hat{p}} = \frac{\text{number of successes}}{\text{total observations}}$$`

- The .hi.kelly[mean] of the sampling distribution is `$p$`
- The .hi.kelly[standard deviation] of the sampling distribution is
`$$\mathit{SE}_p=\sqrt{\frac{p(1-p)}{n}}$$`

---
## Sampling Distribution of a Sample Proportion

.subheader.alice[Binomial Distribution Review]

Say we draw a simple random sample of size `$n$` from a large population that contains `$p$` proportion of successes.  Let `$\coral{\hat{p}}$` be the .hi.purple[sample proportion] of successes,
$$
\coral{\hat{p}} = \frac{\text{number of successes in the sample}}{n}
$$

The .hi.purple[Central Limit Theorem] tells us that with a large enough sample size, the standardized value of `$\coral{\hat{p}}$` will be approximately normal:
$$
\frac{\coral{\hat{p}} - p}{\sqrt{p(1-p)/n}} \sim N(0, 1)
$$

- `$p$` is the true population proportion

- `$\sqrt{p(1-p)/n}$` is the true population standard deviation

---
## Example: Election

A poll by YouGov asked `$1360$` voters in Pennsylvania if they were going to vote for Biden or Trump the day before the election. We will code a vote for Biden `$=1$`, so the proportion `$\coral{\hat{p}}$` is the proportion of people who will vote for Biden. Biden will win Pennsylvania if the population portion is `$p > .5$`.

They find that `$\coral{\hat{p}} = .53$`.

**What is the sampling distrubtion of** `$\coral{\mathbf{\hat{p}}}$`?

### Answer
`$$\coral{\hat{p}} \sim N\left(p_0, \frac{p_0(1-p_0)}{n}\right)$$`
**What's the probability Biden Wins PA?**

Using the sampling distribution, what's the probability that `$p > .50$`?

```r
1-pnorm(q = .50, mean = 0.53, sd = sqrt(0.50*(1-0.50)/1360))
```

```
## [1] 0.9865405
```

.footnote.small[Source: https://projects.fivethirtyeight.com/polls/]

---
## Confidence Intervals for a Population Proportion

We follow the same path from sampling distribution to confidence interval as we did for `$\bar{X}$`

Note, the standard deviation of `$\coral{\hat{p}}$` depends on the parameter `$p$` -- a value that we don't know.

We therefore estimate the standard deviation with the standard error of `$\coral{\hat{p}}$`:

$$ 
SE_{\coral{\hat{p}}}=\sqrt{\frac{\coral{\hat{p}}(1-\coral{\hat{p}})}{n}} 
$$

- Remember! Estimating `$SE$` means `$t$`-dist if sample is small!!!

---
## Confidence Intervals for a Population Proportion

Say we draw a simple random sample of size `$n$` from a large population that contains an unknown proportion `$p$` of successes. An approximate C% .hi.purple[confidence interval] for p is:

$$ 
\coral{\hat{p}} \pm z_{\frac{1-c}{2}} \sqrt{\frac{\coral{\hat{p}}(1-\coral{\hat{p}})}{n}} 
$$

**What do we mean by large?** Can only use this confidence interval when number of successes and failures in the sample are both at least 15 (to remember, half of 30 each).

---
## Example

A poll by YouGov asked 1360 voters in Pennsylvania if they were going to vote for Biden or Trump. We will code a vote for Biden `$=1$`, so the proportion `$\coral{\hat{p}}$` is the proportion of people who will vote for biden. Biden will win Pennsylvania if the population portion is `$p > .5$`.

They find that `$\coral{\hat{p}} = .53$`. What is the sampling distrubtion of `$\coral{\hat{p}}$`?

Check the conditions:

- SRS `$\checkmark$`

- number of success ($1360 * 0.53$) and failures ($1360 * 0.47$) are both larger than `$15$` `$\checkmark$`

So we can go ahead and calculate a 95% confidence interval for the population parameter `$p$`...

---
## Clicker Question
We are given that `$n = 670$`, `$\coral{\hat{p}} = 0.85$`, we will use the standard error of the sample proportion as

$$
  SE_{\coral{\hat{p}}}=\sqrt{\coral{\hat{p}}(1-\coral{\hat{p}})/n}
$$

Which of the following is the correct calculation for a 95% confidence interval?

---
## Hypothesis Testing

We design a hypothesis test such as:

$$ 
H_0: \coral{\hat{p}} = p_0 \ \text{ vs. } \ H_1: \coral{\hat{p}} \neq p_0 
$$

Or one-sided alternatives, such as: `$\coral{\hat{p}} < p_0$` or `$\coral{\hat{p}} > p_0$`.

We reject `$H_0$` if our p-value is lower than our *level of significance*

- p-value: probability of calculating the sample proportion we have, or more extreme value, *given* the null hypothesis is true

---
## Test Statistic

Draw an SRS of size `$n$` from a large population that contains an unknown proportion `$p$` of successes. To test the hypothesis `$H_0: \coral{\hat{p}} = p_0$`, compute the following z-statistic:
$$ 
z=\frac{\coral{\hat{p}}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}
$$

Look up this `$z$` value in the `$z$`-table when the sample size `$n$` is so large that both `$n \cdot p_0$` and `$n \cdot (1-p_0) = 15$` or more.

---
## Example
A survey found that 571 out of 670 (85%) of Americans answered a question on experimental design correctly. Do these data provide convincing evidence that more than 80% of Americans have a good intuition about experimental design?

$$ 
  H_0: p=0.8 \ \text{ vs. } \ H_1: p>0.8 
$$

---
## Example

A survey found that 571 out of 670 (85%) of Americans answered a question on experimental design correctly. Do these data provide convincing evidence that more than 80% of Americans have a good intuition about experimental design?

$$ 
  H_0: p = 0.8 \ \text{ vs. } \ H_1: p > 0.8 
$$

Calculate the the p-value:
$$ 
P(\coral{\hat{p}} > 0.85 \ \vert \ p = 0.8) 
$$

$$ 
P(z > \frac{0.85-0.8}{\sqrt{\frac{0.8 \cdot 0.2}{670}}}) = P(z > 3.25) = 0.0006 
$$

Since  `$p$`-value `$= 0.0006 < \alpha = 0.05$`, reject `$H_0$`.

---
## Practice
On Nov. 1st, the New York Times and Siena College released a poll for Wisconsin with `$n = 1253$` and the sample proportion of people supporting Biden was `$\coral{\hat{p}} = 0.52$`. On election day, we learned the population proportion supporting Biden was `$p = 0.495$`. Would we have rejected the following hypothesis at the `$\alpha = 0.05$` significance level?

$$ 
H_0: p = 0.495
$$

$$ 
H_1: p > 0.495
$$

---
## Election Polling and Simple Random Sample

When do we reject a true null hypothesis?

- .hi.purple[Undercoverage]: when some groups in the population are left out of the process of choosing the sample

- .hi.purple[Oversampling]: when some groups are sampled more often than others in a way that is not representative of the population

- .hi.purple[Nonresponse]: when an individual chosen for the sample can't be contacted or refuses to participate

- .hi.purple[Response Bias]: a systematic pattern of incorrect responses in a sample survey

- .hi.purple[Wording Effect]: a systematic pattern of responses due to poor (or manipulated) wording of survey questions

---
class: center, middle, sydney-blue

#Comparing Two Proportions

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

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      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col" id="Population">Population</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="Pop. Proportion">Pop. Proportion</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="Sample Size">Sample Size</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="Sample Proportion">Sample Proportion</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="Population" class="gt_row gt_right">1</td>
<td headers="Pop. Proportion" class="gt_row gt_left">$ p_1 $</td>
<td headers="Sample Size" class="gt_row gt_left">$ n_1 $</td>
<td headers="Sample Proportion" class="gt_row gt_left">$ \hat{p}_1 $</td></tr>
    <tr><td headers="Population" class="gt_row gt_right">2</td>
<td headers="Pop. Proportion" class="gt_row gt_left">$ p_2 $</td>
<td headers="Sample Size" class="gt_row gt_left">$ n_2 $</td>
<td headers="Sample Proportion" class="gt_row gt_left">$ \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: 
$$ 
\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}} 
$$

---
## Large-Sample Confidence Intervals for Comparing Proportions

Using the equation for standard error:
$$ 
\mathit{SE} = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}} 
$$ 
The confidence interval is constructed as:
$$ 
\hat{p}_1 - \hat{p}_2 \pm z^* \mathit{SE},
$$

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

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

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  vertical-align: middle;
  text-align: left;
}

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  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
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  border-top-color: #D3D3D3;
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  vertical-align: middle;
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#kktelxqgbu .gt_from_md > :first-child {
  margin-top: 0;
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  margin-bottom: 0;
}

#kktelxqgbu .gt_row {
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  text-indent: 25px;
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</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="Population">Population</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="No. Successes">No. Successes</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Sample Size">Sample Size</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Sample Proportion">Sample Proportion</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="Population" class="gt_row gt_center">1</td>
<td headers="No. Successes" class="gt_row gt_center">75</td>
<td headers="Sample Size" class="gt_row gt_center">100</td>
<td headers="Sample Proportion" class="gt_row gt_center">$ \hat{p}_1 $ = 0.75</td></tr>
    <tr><td headers="Population" class="gt_row gt_center">2</td>
<td headers="No. Successes" class="gt_row gt_center">56</td>
<td headers="Sample Size" class="gt_row gt_center">100</td>
<td headers="Sample Proportion" class="gt_row gt_center">$ \hat{p}_2 $ = 0.56</td></tr>
  </tbody>
  
  
</table>
</div>

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

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:

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

---
## Example

<div id="hrlbktovbi" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;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");
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<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="Population">Population</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="No. Successes">No. Successes</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Sample Size">Sample Size</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Sample Proportion">Sample Proportion</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="Population" class="gt_row gt_center">1</td>
<td headers="No. Successes" class="gt_row gt_center">212</td>
<td headers="Sample Size" class="gt_row gt_center">616</td>
<td headers="Sample Proportion" class="gt_row gt_center">$ \hat{p}_1 $ = 0.344</td></tr>
    <tr><td headers="Population" class="gt_row gt_center">2</td>
<td headers="No. Successes" class="gt_row gt_center">7</td>
<td headers="Sample Size" class="gt_row gt_center">49</td>
<td headers="Sample Proportion" 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 `$< \alpha$`