Question 9 options:
In a random sample of 80 people, 52 consider themselves as baseball fans. Compute a 92% confidence interval for the true proportion of people consider themselves as baseball fans and fill in the blanks appropriately.
We are 92% confident that the true proportion of people consider themselves as baseball fans is between

To compute a 92% confidence interval for the true proportion of people who consider themselves as baseball fans, we'll use the following formula for the confidence interval of a proportion:

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

Where:
- \(\hat{p}\) = sample proportion (the proportion of people in the sample who consider themselves as baseball fans)
- \(z_{\alpha/2}\) = the critical value corresponding to the desired confidence level (92%)
- \(n\) = sample size

### Step 1: Calculate \(\hat{p}\)

\[
\hat{p} = \frac{52}{80} = 0.65
\]

### Step 2: Find the critical value \(z_{\alpha/2}\)

For a 92% confidence level, the area in both tails of the normal distribution is \(1 - 0.92 = 0.08\), so each tail has an area of 0.04. Using a z-table, we find that the corresponding critical value \(z_{\alpha/2}\) is approximately **1.75**.

### Step 3: Compute the margin of error

\[
\text{Margin of error} = z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
\]

Substituting the known values:

\[
\text{Margin of error} = 1.75 \times \sqrt{\frac{0.65(1 - 0.65)}{80}} = 1.75 \times \sqrt{\frac{0.65 \times 0.35}{80}}
\]

Let me compute this.The 92% confidence interval for the true proportion of people who consider themselves baseball fans is approximately between **0.557** and **0.743**.

Thus, we are 92% confident that the true proportion of people who consider themselves baseball fans is between **55.7% and 74.3%**.

Do you have any questions or need further details? 

Here are five related questions to expand your understanding:
1. How would the confidence interval change if we increased the confidence level to 95%?
2. What happens to the confidence interval if the sample size increases?
3. How do you interpret the margin of error in this context?
4. What assumptions do we make when constructing a confidence interval for a proportion?
5. How would the confidence interval change if the sample proportion were 0.75 instead of 0.65?

**Tip:** Increasing the confidence level makes the interval wider because we need to account for more uncertainty at higher confidence levels.

In a random sample of 80 people, 52 consider themselves as baseball fans. Compute a 92% confidence interval for the true proportion of people consider themselves as baseball fans and fill in the blanks appropriately. We are 92% confident that the true proportion of people consider themselves as baseball fans is between ____.

Calculate a 92% confidence interval for the proportion of baseball fans in a sample of 80 people. The sample proportion is 0.65, and the confidence interval is between 55.7% and 74.3%. This problem covers key concepts like confidence intervals, proportions, and the use of critical values.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation