https://file.aiquickdraw.com/mathbot/file/172633436966047j1ste2.jpg

The problem asks you to calculate the probability of committing a Type II error (\(\beta\)) for a hypothesis test involving the following information:

- Null hypothesis (\(H_0\)): \(\mu \leq 30\)
- Alternative hypothesis (\(H_1\)): \(\mu > 30\)
- Standard deviation (\(\sigma\)): 7
- Sample size (\(n\)): 40
- Significance level (\(\alpha\)): 0.05
- Actual mean (\(\mu = 33\))

Let's break down the steps to calculate \(\beta\) (the probability of a Type II error):

### Steps to calculate \(\beta\):
1. **Find the critical value for the test**: This is based on the significance level \(\alpha = 0.05\) for a one-tailed test.
2. **Determine the test statistic (z-score) for the critical value**.
3. **Calculate the non-centrality parameter**: The distance between the hypothesized mean (\( \mu_0 = 30\)) and the true mean (\(\mu = 33\)).
4. **Find the probability of making a Type II error (\(\beta\)) using the non-central z-distribution**.

I'll start by calculating the critical z-value and proceed with the computations.The probability of committing a Type II error (\(\beta\)) is approximately **0.1433**.

This means that there is a 14.33% chance of failing to reject the null hypothesis when the true mean is actually \(\mu = 33\).

Would you like more details on any of the steps involved? Here are five related questions for further exploration:

1. What is the difference between a Type I and Type II error?
2. How does the sample size affect the probability of a Type II error?
3. What is the role of the critical value in hypothesis testing?
4. How does changing the significance level (\(\alpha\)) influence \(\beta\)?
5. How would you calculate the power of the test?

**Tip:** To decrease the probability of a Type II error, consider increasing the sample size or adjusting the significance level.

Given α = 0.05, n = 40, σ = 7, calculate β for μ = 33 in a hypothesis test where H0: μ ≤ 30 and H1: μ > 30.

Learn how to calculate the probability of committing a Type II error (β) for a hypothesis test involving a null hypothesis μ ≤ 30 and an alternative hypothesis μ > 30, given a sample size n = 40, α = 0.05, and standard deviation σ = 7. This problem involves hypothesis testing, Z-scores, and the central limit theorem.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation