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 $\alpha =0.05$ level, can it be concluded that the mean rainfall was below the reported average? What if $\alpha =0.01$? Assume the amount of summer rainfall follows a normal distribution.

Short answer

There is sufficient evidence to conclude that mean rainfall was below the reported average$11.52$inches.

Step-by-step solution

Given information

$\text{M}=7.42\text{inches},\text{S}=1.3\text{inches},\text{n}=10$.

Explanation

State the hypothesis:

The null hypothesis states that mean rainfall is greater than or equal to the reported average $11.52$inches. In symbols:

$H_0:\mu \ge 11.52$

The alternative hypothesis states that mean rainfall is less than the reported average $11.52$inches. In symbols:

$H_a:\mu <11.52$

The degree of freedom is:

$$\begin{gathered} df=n-1 \\ =10-1 \\ =9 \end{gathered}$$

The test statistic is:

$$\begin{gathered} t=\frac{M-\mu }{S/\sqrt{n}} \\ =\frac{7.42-11.52}{1.3/\sqrt{10}} \\ =\frac{-4.10}{0.4111} \\ =-9.97 \end{gathered}$$

The $p-valueis0.0000$

Since $p-value<0.05$and $p-value<0.01$also, reject the null hypothesis at $a=0.05anda=0.01$both. We conclude that mean rainfall is less than the reported average $11.52$inches.