Question

The Weather Underground reported that the mean amount of summer rainfall for the northeastern US is at least 11.52 inches. Ten cities in the northeast are randomly selected and the mean rainfall amount is calculated to be 7.42 inches with a standard deviation of 1.3 inches. At the α = 0.05 level, can it be concluded that the mean rainfall was below the reported average? What if α = 0.01? Assume the amount of summer rainfall follows a normal distribution.

a) Yes, at α = 0.05, it can be concluded that the mean rainfall was below the reported average.

b) No, at α = 0.05, it cannot be concluded that the mean rainfall was below the reported average.

c) Yes, at α = 0.01, it can be concluded that the mean rainfall was below the reported average.

d) No, at α = 0.01, it cannot be concluded that the mean rainfall was below the reported average.

Answer 1

Using hypothesis testing, we can conclude that the mean rainfall in the northeastern US is below the reported average, both at alpha = 0.05 and alpha = 0.01 levels.

Step-by-step explanation:

To determine whether it can be concluded that the mean rainfall was below the reported average, we need to perform a hypothesis test.

1. Null hypothesis: The mean rainfall is equal to or greater than 11.52 inches.
2. Alternative hypothesis: The mean rainfall is less than 11.52 inches.
3. Calculate the test statistic using the formula: test statistic = (sample mean - population mean) / (standard deviation / sqrt(sample size)) = (7.42 - 11.52) / (1.3 / sqrt(10)) = -12.12.
4. Find the critical value corresponding to alpha = 0.05 from the t-distribution table with 9 degrees of freedom. The critical value is -1.833.
5. Since the test statistic (-12.12) is less than the critical value (-1.833), we reject the null hypothesis.
6. At alpha = 0.05 level, we conclude that the mean rainfall was below the reported average.
7. In the case of alpha = 0.01, the critical value from the t-distribution table with 9 degrees of freedom is -2.821. The test statistic (-12.12) is still less than the critical value, so we reject the null hypothesis at alpha = 0.01 level as well.
8. Hence, the correct answer is c) Yes, at α = 0.01, it can be concluded that the mean rainfall was below the reported average.