Textbook Problem

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consider the following hypothesis test.

 H₀: μ ≥ 10

 H_a: μ < 10

 The sample size is 120 and the population standard deviation is assumed known with σ = 5. Use

 α = .05.

1. a. If the population mean is 9, what is the probability that the sample mean leads to the conclusion do not reject H₀?
2. b. What type of error would be made if the actual population mean is 9 and we conclude that H₀: μ ≥ 10 is true?
3. c. What is the probability of making a Type II error if the actual population mean is 8?

To determine

Find the probability that the sample mean leads to the conclusion do not reject $H_0$.

Explanation of Solution

Calculation:

The given information is that the sample size is $n=120$, the population standard deviation is $\sigma =5$, $\mu =9$ and $\alpha =0.05$.

The hypotheses are given below:

Null hypothesis:

$H_0:\mu \ge 10$

Alternative hypothesis:

$H_a:\mu <10$

The rejection rule for lower tail test using the critical value is,

If $z\le -z_{\alpha }$, reject the null hypothesis

Critical value:

Software Procedure:

Step-by-step procedure to obtain the critical value using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability> OK.
• From Distribution, choose ‘Normal’ distribution.
• Click the Shaded Area tab.
• Choose Probability Value and Left Tail for the region of the curve to shade.
• Enter the Probability value as 0.05.
• Click OK.

Output using the MINITAB software is given below:

From the output, the critical value is −1.645.

Rejection rule:

If $z\le -1.645$, reject the null hypothesis

The value $\overline{x}$ is,

$\begin{array}{c}\overline{x}=\mu -z\left(\frac{\sigma }{\sqrt{n}}\right)\\ =10-1\end{array}$

To determine

Identify the type of error would be made if the actual population mean is 9 and conclude that $H_0:\mu \ge 10$ is true.

To determine

Find the probability of making a Type II error if the actual population mean is 8.