$9.1-9.2$ Question 6 of 13 possible This question: 1 Submit quiz point(s) possible The following data represent the $\mathrm{pH}$ of rain for a random sample of 12 rain dates. A normal probability plot suggests the data could come from a population that is normally distributed. A boxplot indicates there are no outliers. Complete parts a) through d) below. 5.20 5.72 4.89 4.80 5.02 4.57 4.74 5.19 5.29 4.76 4.56 5.69 Click the icon to view the table of critical t-values. (Round to two decimal places as r.coded.) (b) Construct and interpret a $95 \%$ confidence interval for the mean pH of rainwater. Select the correct choice below and fill in the answer boxes to complete your choice. (Use ascending order. Round to two decimal places as needed.) A. If -repeated samples are taken, $95 \%$ of them will have a sample $\mathrm{pH}$ of rain water between $\square$ and $\square$. B. There is $95 \%$ confidence that the population mean $\mathrm{pH}$ of rain water is between $\square$ and $\square$. C. There is a $95 \%$ probability that the true mean $\mathrm{pH}$ of rain water is between $\square$ and $\square$.
Solution
Step 1 ：Given the data of pH of rainwater for 12 random samples: 5.20, 5.72, 4.89, 4.80, 5.02, 4.57, 4.74, 5.19, 5.29, 4.76, 4.56, 5.69.
Step 2 ：Calculate the sample mean (\(\bar{x}\)) and the sample standard deviation (s). The sample mean is 5.035833333333333 and the sample standard deviation is 0.3926010775082082.
Step 3 ：Calculate the t-score for a two-tailed test with \(\alpha = 0.05\) and \(n-1\) degrees of freedom. The t-score is 2.200985160082949.
Step 4 ：Use the formula for a confidence interval: \(\bar{x} \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}\).
Step 5 ：Substitute the values into the formula to get the confidence interval: (4.786386509472255, 5.285280157194411).
Step 6 ：Round the confidence interval to two decimal places to get: (4.79, 5.29).
Step 7 ：Final Answer: The 95% confidence interval for the mean pH of rainwater is between 4.79 and 5.29. Therefore, there is 95% confidence that the population mean pH of rain water is between \(\boxed{4.79}\) and \(\boxed{5.29}\).
