Question

you are given the sample mean and the population standard deviation. use this information to construct the 90% and 95% confidence intervals for the population mean. interpret the results and compare the widths of the confidence intervals.
from a random sample of 47 business days, the mean closing price of a certain stock was $116.50. assume the population standard deviation is $9.71.
the 90% confidence interval is (114.17, 118.83).
(round to two decimal places as needed.)
the 95% confidence interval is (□,□).
(round to two decimal places as needed.)

Explanation:

Step1: Recall the formula for confidence interval

The formula for a confidence interval when the population standard deviation \(\sigma\) is known is \(\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(z_{\alpha/2}\) is the z - score, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.

Step2: Find the z - score for 95% confidence interval

For a 95% confidence interval, \(\alpha = 1 - 0.95=0.05\), and \(\alpha/2 = 0.025\). From the standard normal table, \(z_{0.025}=1.96\).

Step3: Identify the values of \(\bar{x}\), \(\sigma\), and \(n\)

Given \(\bar{x}=116.50\), \(\sigma = 9.71\), and \(n = 47\).

Step4: Calculate the margin of error \(E\)

\(E=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}=1.96\times\frac{9.71}{\sqrt{47}}\). First, \(\sqrt{47}\approx6.86\), then \(\frac{9.71}{6.86}\approx1.42\), and \(1.96\times1.42 = 2.78\).

Step5: Calculate the confidence interval

The lower limit is \(\bar{x}-E=116.50 - 2.78=113.72\), and the upper limit is \(\bar{x}+E=116.50 + 2.78=119.28\).

Answer:

The 95% confidence interval is \((113.72,119.28)\)