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
5.30
5.02
5.72
4.38
4.80
4.61
4.57
4.74
5.19
4.56
5.68 there are no outliers. Complete parts a) through d) below.
Click the icon to view the table of critical t-values.
(a) Determine a point estimate for the population mean.
A point estimate for the population mean is 4.94 . (Round to two decimal places as needed.)
(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. 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 and

C. If repeated samples are taken, $95 \%$ of them will have a sample $\mathrm{pH}$ of rain water between and

Step 1 ：Given the data set of pH values of rainwater, we are asked to find the point estimate for the population mean and a 95% confidence interval for the mean pH of rainwater.

Step 2 ：The point estimate for the population mean is simply the sample mean, which can be calculated by summing all the values and dividing by the number of values.

Step 3 ：The 95% confidence interval for the mean pH of rainwater can be calculated using the formula for a confidence interval: \(\bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(t_{\alpha/2}\) is the t-value for a 95% confidence interval, \(s\) is the sample standard deviation, and \(n\) is the sample size.

Step 4 ：Using the given data, we calculate the sample mean to be approximately 4.96.

Step 5 ：We also calculate the sample standard deviation to be approximately 0.46.

Step 6 ：The sample size is 11.

Step 7 ：Using a t-distribution table, we find that the t-value for a 95% confidence interval with 10 degrees of freedom (n-1) is approximately 2.23.

Step 8 ：Substituting these values into the confidence interval formula, we find the 95% confidence interval for the mean pH of rainwater to be approximately between 4.65 and 5.27.

Step 9 ：Final Answer: The point estimate for the population mean is \(\boxed{4.96}\). There is 95% confidence that the population mean pH of rain water is between \(\boxed{4.65}\) and \(\boxed{5.27}\).