What proportion of commercial airline pilots are more than 40 years of age? Suppose a researcher has access to a list of all pilots who are members of the Commercial Airline Pilots Association. If this list is used as a frame for the study, she can randomly select a sample of pilots, contact them, and ascertain their ages. From 89 of these pilots so selected, she learns that 48 are more than 40 years of age. Construct an 85% confidence interval to estimate the population proportion of commercial airline pilots who are more than 40 years of age. What is the upper estimate value of the population proportion?

Round the answers to 3 decimal places.


To construct an 85% confidence interval for the population proportion of commercial airline pilots who are more than 40 years of age, we will use the following formula:

\[
\hat{p} \pm z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]

Where:
- \(\hat{p}\) is the sample proportion.
- \(z_{\alpha/2}\) is the critical value for an 85% confidence interval.
- \(n\) is the sample size.

### Step 1: Calculate the sample proportion (\(\hat{p}\))
\[
\hat{p} = \frac{x}{n} = \frac{48}{89} \approx 0.539
\]

### Step 2: Find the critical value (\(z_{\alpha/2}\))
For an 85% confidence interval, the corresponding critical value can be found using standard normal distribution tables or a calculator. For 85% confidence, \(z_{\alpha/2} = 1.440\).

### Step 3: Calculate the margin of error
\[
\text{Margin of Error} = z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = 1.440 \times \sqrt{\frac{0.539(1 - 0.539)}{89}}
\]

Now, let me compute this.The sample proportion \(\hat{p}\) is approximately 0.539, and the margin of error is 0.076. 

Thus, the 85% confidence interval for the population proportion of commercial airline pilots who are more than 40 years of age is:

\[
(0.463, 0.615)
\]

The **upper estimate** of the population proportion is approximately **0.615** (rounded to three decimal places).

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you find the critical value for a different confidence level?
2. What happens to the confidence interval if the sample size increases?
3. How would you interpret the margin of error in this context?
4. How would a 95% confidence interval differ from an 85% one?
5. What is the impact of increasing the sample proportion on the interval width?

**Tip:** The larger the sample size, the smaller your confidence interval will be, leading to more precise estimates.

What proportion of commercial airline pilots are more than 40 years of age? Suppose a researcher has access to a list of all pilots who are members of the Commercial Airline Pilots Association. If this list is used as a frame for the study, she can randomly select a sample of pilots, contact them, and ascertain their ages. From 89 of these pilots so selected, she learns that 48 are more than 40 years of age. Construct an 85% confidence interval to estimate the population proportion of commercial airline pilots who are more than 40 years of age. What is the upper estimate value of the population proportion?

Learn how to calculate an 85% confidence interval to estimate the population proportion of commercial airline pilots over 40 years of age. From a sample of 89 pilots, 48 were over 40. Find the upper estimate of the population proportion with a detailed, step-by-step solution using key statistical formulas.

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