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.

Short answer

At the 5% significance level, there is enough evidence to conclude that the mean amount of summer rain in the northeaster US is less than 11.52 inches, on average.

Step-by-step solution

state the null and alternate hypothesis. we have to conduct hypothesis test if the mean amount of summer rainfall for the northeastern US takes less than 11.52 inches of rainfall according to Weather underground report.

$H_0:\mu \ge 11.52;H_a:\mu <11.52$

Calculate the p-value using the normal distribution for proportions: 

$p-value=0.000002$which is almost 0.

In one to two complete sentences, explain what the p-value means for this problem. If the null hypothesis is true (the mean rainfall is 11.52inches), then there is a 0.000002 probability that the sample (estimated) mean rainfall amount is 7.42inches or more.

Compare α and the p-value:Indicate the correct decision (“reject” or “do not reject” the null hypothesis), the reason for it, and write an appropriate conclusion, using complete sentences.

alpha	decision	reason for decision
0.01	Reject the null hypothesis.	$p-value<0.01$

Conclusion: At the 5% significance level, there is enough evidence to conclude that the mean amount of summer rain in the northeastern US is less than 11.52 inches, on average. We would make the same conclusion if alpha was 1% because the p-value is almost 0.