Chapter SH, Problem 11.14EP

a.

To determine

To state the claim and hypothesis and draw the conclusion.

Explanation of Solution

Given:

The number of students that run for students council is normally distributed with a mean of $16.8$ and a standard deviation of $3.7$ .

Calculation:

Hypothesis test is used to make a specific claim about the mean. The null hypothesis denoted by $H_0$ is a statement of equality used to be tested and the alternative hypothesis denoted by $H_a$ is the statement of equality that it is the compliment of the null hypothesis.

Here, $H_a:\mu <100$ , the hypothesis test is a left tailed test.

First step:

State the claim and hypothesis,

  $\begin{array}{l}{H}_0:\mu =100\text{ (claim)}\\ H_a:\mu <100\end{array}$

Second step:

Determine the critical region,

The alternative hypothesis is $\mu <100$ , so, this is a left tailed test. Now, the user is testing at $5\%$ significance. So, the user needs to identify the z-value that corresponds with the lower $5\%$ of distribution.

Third step:

Calculate the z-statistics for the sample data; the formula is given below;

  $z=\frac{x-\mu }{\frac{s}{\sqrt{n}}}$

Substituting the values:

  $\begin{array}{l}z=\frac{98.7-100}{\frac{1.512}{\sqrt{30}}}\\ z=\frac{-1.3}{\frac{1.512}{5.4}}\\ z=-4.71\end{array}$

Fourth step:

  $H_0$ is rejected because the z-statistic for the sample falls neither in the critical region.

Fifth step:

Make a conclusion about the claim;

There is enough evidence to reject the claim. Thus, every box of cereal contains $100$ raisins.