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.
\begin{tabular}{llll}
5.20 & 5.72 & 4.89 & 4.80 \\
5.02 & 4.57 & 4.74 & 5.19 \\
4.87 & 4.76 & 4.56 & 5.70
\end{tabular}
Click the icon to view the table of critical t-values.
(Round to two decimal places as needed.)
(b) Construct and interpret a $95 \%$ confidence interval for the mean $\mathrm{pH}$ 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 $95 \%$ confidence that the population mean $\mathrm{pH}$ of rain water is between $\square$ and $\square$.
B. If repeated samples are taken, $95 \%$ of them will have a sample $\mathrm{pH}$ of rain water between $\square$ and
C. There is a $95 \%$ probability that the true mean $\mathrm{pH}$ of rain water is between $\square$ and $\square$.
Answer
Thus, we can say that there is 95% confidence that the population mean pH of rain water is between \(\boxed{4.76}\) and \(\boxed{5.25}\).
Step 1 ：Given the data set of pH values of rainwater, we are asked to construct a 95% confidence interval for the mean pH. The data set is [5.2, 5.72, 4.89, 4.8, 5.02, 4.57, 4.74, 5.19, 4.87, 4.76, 4.56, 5.7].
Step 2 ：First, we calculate the sample mean and the sample standard deviation. The sample mean is calculated as the sum of all the values divided by the number of values. The sample standard deviation is a measure of the amount of variation or dispersion in the set of values.
Step 3 ：The sample mean (\(\bar{x}\)) is approximately 5.00 and the sample standard deviation (s) is approximately 0.39.
Step 4 ：The sample size (n) is 12, so the degrees of freedom (df) is n - 1 = 11.
Step 5 ：We then find the t-score corresponding to a 95% confidence level with 11 degrees of freedom. The t-score (t) is approximately 2.20.
Step 6 ：We can now calculate the confidence interval using the formula \(\bar{x} \pm t \cdot \frac{s}{\sqrt{n}}\).
Step 7 ：The lower bound of the confidence interval is approximately 4.76 and the upper bound is approximately 5.25.
Step 8 ：Thus, we can say that there is 95% confidence that the population mean pH of rain water is between \(\boxed{4.76}\) and \(\boxed{5.25}\).
