Math Problem Statement
Solution
The problem asks you to calculate the probability of committing a Type II error () for a hypothesis test involving the following information:
- Null hypothesis ():
- Alternative hypothesis ():
- Standard deviation (): 7
- Sample size (): 40
- Significance level (): 0.05
- Actual mean ()
Let's break down the steps to calculate (the probability of a Type II error):
Steps to calculate :
- Find the critical value for the test: This is based on the significance level for a one-tailed test.
- Determine the test statistic (z-score) for the critical value.
- Calculate the non-centrality parameter: The distance between the hypothesized mean () and the true mean ().
- Find the probability of making a Type II error () 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 () 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 .
Would you like more details on any of the steps involved? Here are five related questions for further exploration:
- What is the difference between a Type I and Type II error?
- How does the sample size affect the probability of a Type II error?
- What is the role of the critical value in hypothesis testing?
- How does changing the significance level () influence ?
- 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.
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Math Problem Analysis
Mathematical Concepts
Hypothesis Testing
Type II Error (β)
Z-scores
Normal Distribution
Formulas
Z = (X̄ - μ) / (σ / √n)
Type II Error (β) is the probability of failing to reject H₀ when H₁ is true.
Theorems
Central Limit Theorem
Suitable Grade Level
Undergraduate statistics or advanced high school level
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