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 \(\boxed{5.25}\).