Questions: Question 23 of 23 (1 point) Question Attempt 1 of Unlimited Alternate Answer: 0.44 Part 2 of (b) Construct a 99.5% confidence interval for the proportion of all entering freshmen at this college who scored less than 610 on the math SAT. Round the answer to at least three decimal places. A 99.5% confidence interval for the proportion of all entering freshmen at this college who scored less than 610 on the math SAT is <p<

Solution Steps

We are tasked with constructing a \(99.5\%\) confidence interval for the proportion of all entering freshmen at a college who scored less than \(610\) on the math SAT. The sample proportion (\(\hat{p}\)) is given as \(0.44\), and we assume a sample size (\(n\)) of \(100\).

For a \(99.5\%\) confidence level, the significance level (\(\alpha\)) is: \[ \alpha = 1 - 0.995 = 0.005 \] The critical value \(z\) corresponding to \(\alpha/2 = 0.0025\) can be found using the standard normal distribution. For this confidence level, \(z \approx 2.807\).

The standard error (SE) for the sample proportion is calculated using the formula: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.44(1 - 0.44)}{100}} = \sqrt{\frac{0.44 \cdot 0.56}{100}} = \sqrt{0.2464 \cdot 0.01} = \sqrt{0.002464} \approx 0.0496 \]

The confidence interval is calculated using the formula: \[ \hat{p} \pm z \cdot SE \] Substituting the values: \[ 0.44 \pm 2.807 \cdot 0.0496 \] Calculating the margin of error: \[ 2.807 \cdot 0.0496 \approx 0.139 \] Thus, the confidence interval is: \[ (0.44 - 0.139, 0.44 + 0.139) = (0.301, 0.579) \]

Final Answer