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 $\begin{array}{llll}5.30 & 5.72 & 4.38 & 4.80 \\ 5.02 & 4.57 & 4.74 & 5.19 \\ 4.61 & 4.76 & 4.56 & 5.68\end{array}$ normally distributed. A boxplot indicates there are no outliers. Complete parts a) through d) below. Click the icon to view the table of critical t-values. A. There is a $95 \%$ probability that the true mean $\mathrm{pH}$ of rain water is between and B. There is $95 \%$ confidence that the population mean $\mathrm{pH}$ of rain water is between 4.65 and 5.23 '. C. If repeated samples are taken, $95 \%$ of them will have a sample $\mathrm{pH}$ of rain water between and (c) Cornstruct and interpret a 99\% 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. There is $99 \%$ confidence that the population mean $\mathrm{pH}$ of rain water is between and B. If repeated samples are taken, $99 \%$ of them will have a sample $\mathrm{pH}$ of rain water between and C. There is a $99 \%$ probability that the true mean $\mathrm{pH}$ of rain water is between and
Solution
Step 1 ：Given the data of pH values of rainwater, we are asked to find the 99% confidence interval for the mean pH of rainwater.
Step 2 ：The data is: \(5.3, 5.72, 4.38, 4.8, 5.02, 4.57, 4.74, 5.19, 4.61, 4.76, 4.56, 5.68\).
Step 3 ：The sample mean \(\bar{x}\) is calculated to be approximately 4.944.
Step 4 ：The sample standard deviation \(s\) is calculated to be approximately 0.442.
Step 5 ：The sample size \(n\) is 12.
Step 6 ：The critical value \(t_{\alpha/2}\) for a 99% confidence interval with 11 degrees of freedom (since \(n-1 = 12-1 = 11\)) is approximately 3.106.
Step 7 ：We can calculate the confidence interval using the formula: \[\bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}\]
Step 8 ：Substituting the values into the formula, we get the confidence interval to be approximately \((4.548, 5.341)\).
Step 9 ：\(\boxed{\text{There is a 99\% confidence that the population mean pH of rain water is between 4.55 and 5.34.}}\)
