Question

you are given the sample mean and the population standard deviation. use this information to construct the 80% and 95% confidence intervals for the population mean. interpret the results and compare the widths of the confidence intervals.
with a random sample of 47 business days, the mean closing price of a certain stock was $116.50. assume the population standard deviation is $0.71.
the 80% confidence interval is (□,□).
(round to two decimal places as needed.)

Explanation:

Step1: Find the z - score for 80% confidence level

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

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

The formula for the margin of error when the population standard deviation \(\sigma\) is known is \(E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\). Given \(n = 47\), \(\sigma=8.71\), and \(z_{\alpha/2}=1.28\)

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Step3: Calculate the confidence interval

The formula for the confidence interval is \(\bar{x}-E<\mu<\bar{x} + E\). Given \(\bar{x}=116.50\)

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Answer:

\((114.87,118.13)\)