Question

high school graduation gifts a random survey of 24 couples who had children graduating from high school found that they spent an average of $106.98 on graduation gifts. the standard deviation of the sample is $5.20. find the 95% confidence interval for the mean amount of money couples spent for graduation gifts. round intermediate calculations to three decimal places, and final answers to two decimal places.
$ 104.79 < μ < $ 109.18

Explanation:

Step1: Find the degrees of freedom and critical value

The degrees of freedom \(df=n - 1=24-1 = 23\). For a 95% confidence interval, \(\alpha=1 - 0.95=0.05\), and \(\frac{\alpha}{2}=0.025\). Using the t - distribution table or a calculator, the critical value \(t_{\frac{\alpha}{2},df}=t_{0.025,23}=2.069\) (rounded to three decimal places).

Step2: Calculate the margin of error

The formula for the margin of error \(E=t_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}\). Given \(s = 5.20\), \(n = 24\), and \(t_{\frac{\alpha}{2}}=2.069\).

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Step3: Calculate the confidence interval

The confidence interval for the population mean \(\mu\) is \(\bar{x}-E<\mu<\bar{x}+E\). Given \(\bar{x}=106.98\) and \(E = 2.196\).

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Answer:

\(104.78<\mu<109.18\)