Math Problem Statement
Solution
To determine the Type II error () for the alternative hypothesis , we need to understand the components given in the choices:
- Type II error () is the probability of failing to reject the null hypothesis when the alternative hypothesis is true.
- In a two-tailed test (as suggests), the calculation of involves both tails of the distribution because we are concerned with deviations in both directions.
Analysis of Options:
Let's go through the options and see which one fits the setup for in a two-tailed test:
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Option A: This option uses a one-tailed z-score for (typically used in a one-tailed test scenario). This does not match the two-tailed setup.
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Option B: This option also seems aligned with a one-tailed test setup, with a single z-score adjustment rather than the two-tailed approach.
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Option C: This option uses , which is appropriate for a two-tailed test, and subtracts the probabilities on both sides to compute . This makes Option C correct, as it accounts for the two-tailed nature of the test by incorporating .
Correct Answer:
Option C is correct because it accurately represents the Type II error for a two-tailed test when the alternative hypothesis is .
Would you like a further breakdown of how is calculated or more examples on Type I and Type II errors?
Here are some follow-up questions to deepen understanding:
- What is the difference between Type I and Type II errors?
- How do you interpret the significance level () in hypothesis testing?
- Why is it essential to distinguish between one-tailed and two-tailed tests?
- How does the sample size affect the Type II error ()?
- How would you calculate power () in this scenario?
Tip: In hypothesis testing, carefully distinguish between one-tailed and two-tailed tests to choose the appropriate critical values for and .
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Math Problem Analysis
Mathematical Concepts
Hypothesis Testing
Type II Error
Two-Tailed Test
Normal Distribution
Formulas
Type II Error (\( \beta \)) for a two-tailed test: \( \Phi \left( z_{\alpha/2} + \frac{\mu_0 - \mu'}{\sigma / \sqrt{n}} \right) - \Phi \left( -z_{\alpha/2} + \frac{\mu_0 - \mu'}{\sigma / \sqrt{n}} \right) \)
Standard Normal Distribution: \( \Phi(z) \)
Theorems
Central Limit Theorem
Normal Approximation
Suitable Grade Level
Undergraduate Level (Statistics)
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