Question

sat scores: a college admissions officer takes a simple random sample of 100 entering freshmen and computes their mean mathematics sat score to be 470. assume the population standard deviation is \\( \sigma = 116 \\).
(a) construct a 80% confidence interval for the mean mathematics sat score for the entering freshman class. round the answer to the nearest whole number.
a 80% confidence interval for the mean mathematics sat score is \\( 443 < \mu < 473 \\).

Explanation:

Step1: Find the z - score

For an 80% confidence interval, the significance level \(\alpha=1 - 0.80=0.20\). Then \(\alpha/2=0.10\). Looking up in the standard normal distribution table, \(z_{\alpha/2}=z_{0.10}\approx1.28\).

Step2: Calculate the margin of error \(E\)

The formula for the margin of error for a confidence interval of the population mean when the population standard deviation \(\sigma\) is known is \(E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\). Given \(\sigma = 116\), \(n = 100\), and \(z_{\alpha/2}=1.28\). Then \(E=1.28\times\frac{116}{\sqrt{100}}=1.28\times11.6 = 14.848\).

Step3: Construct the confidence interval

The confidence interval is \(\bar{x}-E<\mu<\bar{x} + E\). Given \(\bar{x}=470\). So \(470-14.848<\mu<470 + 14.848\), which is \(455<\mu<485\) (rounded to the nearest whole number).

Answer:

\(455<\mu<485\)