nent 23: Test 4 Review Question 6, 7.1.29-T HW Score: 98\%, 19.6 of 20 points Part 1 of 2 Points: 0.6 of 1 Save Refer to the accompanying data set of 20 randomly selected presidents. Treat the data as a sample and find the proportion of presidents who were taller than their opponents. Use that result to construct a $$ 95 \% $$ confidence interval estimate of the population percentage. Based on the result, does it appear that greater height is an advantage for presidential candidates? Why or why not? Click the icon to view the table of heights. Construct a 95\% confidence interval estimate of the percentage of presidents who were taller than their opponents. $$ \square $$ $$ \%

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To solve this problem, we need to follow these steps:

1. **Extract Known Conditions**: We have a sample of 20 randomly selected presidents. We need to find the proportion of these presidents who were taller than their opponents and then construct a 95% confidence interval for the population percentage.

2. **Calculate the Proportion**:
   - Let $$ n $$ be the total number of presidents in the sample, which is 20.
   - Let $$ x $$ be the number of presidents who were taller than their opponents. We need to find this value from the provided data set.

3. **Construct the Confidence Interval**:
   - The formula for the confidence interval for a proportion is given by:
     $$
     \hat{p} \pm z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
     $$
   - Where:
     - $$ \hat{p} = \frac{x}{n} $$ is the sample proportion.
     - $$ z $$ is the z-value corresponding to the desired confidence level (for 95%, $$ z \approx 1.96 $$).
     - $$ n $$ is the sample size.

4. **Interpret the Results**: After calculating the confidence interval, we will analyze whether greater height appears to be an advantage for presidential candidates based on the results.

### Step 1: Calculate the Proportion

Assuming we have the data from the table of heights, let's denote $$ x $$ as the number of presidents taller than their opponents.

For example, if we find that 12 out of 20 presidents were taller than their opponents, then:
$$
\hat{p} = \frac{x}{n} = \frac{12}{20} = 0.6
$$

### Step 2: Construct the Confidence Interval

Now, we can calculate the confidence interval using the formula provided. Let's calculate it step by step.

1. Calculate $$ \hat{p} $$:
   $$
   \hat{p} = \frac{x}{n}
   $$

2. Calculate the standard error (SE):
   $$
   SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
   $$

3. Calculate the margin of error (ME):
   $$
   ME = z \cdot SE
   $$

4. Construct the confidence interval:
   $$
   \text{CI} = \hat{p} \pm ME
   $$

### Step 3: Execute the Calculations

Let's assume $$ x = 12 $$ for this example. We will calculate the confidence interval based on this assumption.

Now, I will perform the calculations.

1. Calculate $$ \hat{p} $$:
   $$
   \hat{p} = \frac{12}{20} = 0.6
   $$

2. Calculate the standard error (SE):
   $$
   SE = \sqrt{\frac{0.6(1 - 0.6)}{20}} = \sqrt{\frac{0.6 \cdot 0.4}{20}} = \sqrt{\frac{0.24}{20}} = \sqrt{0.012} \approx 0.1095
   $$

3. Calculate the margin of error (ME):
   $$
   ME = 1.96 \cdot 0.1095 \approx 0.214
   $$

4. Construct the confidence interval:
   $$
   \text{CI} = 0.6 \pm 0.214
   $$
   - Lower limit: $$ 0.6 - 0.214 \approx 0.386 $$
   - Upper limit: $$ 0.6 + 0.214 \approx 0.814 $$

### Final Confidence Interval

Converting the proportions to percentages:
- Lower limit: $$ 0.386 \times 100 \approx 38.6\% $$
- Upper limit: $$ 0.814 \times 100 \approx 81.4\% $$

Thus, the 95% confidence interval estimate of the percentage of presidents who were taller than their opponents is:
$$
38.6\% < p < 81.4\%
$$

### Interpretation

Based on this confidence interval, if the majority of the interval is above 50%, it suggests that greater height may be an advantage for presidential candidates. However, if the interval includes values below 50%, it indicates that height may not be a significant factor.

Please provide the actual number of presidents taller than their opponents if you have that data, and I can adjust the calculations accordingly.
The 95% confidence interval for the percentage of presidents who were taller than their opponents is 38.6% to 81.4%.

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