Math Problem Statement

If the alternative hypothesis is \( \mu \neq 5 \), what will be the Type II error (\( \beta \))?

Solution

To determine the Type II error (β\beta) for the alternative hypothesis μ5\mu \neq 5, we need to understand the components given in the choices:

  1. Type II error (β\beta) is the probability of failing to reject the null hypothesis when the alternative hypothesis is true.
  2. In a two-tailed test (as μ5\mu \neq 5 suggests), the calculation of β\beta 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 β\beta in a two-tailed test:

  • Option A: This option uses a one-tailed z-score for α\alpha (typically used in a one-tailed test scenario). This does not match the two-tailed setup.

  • 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.

  • Option C: This option uses zα/2z_{\alpha/2}, which is appropriate for a two-tailed test, and subtracts the probabilities on both sides to compute β\beta. This makes Option C correct, as it accounts for the two-tailed nature of the test by incorporating ±zα/2\pm z_{\alpha/2}.

Correct Answer:

Option C is correct because it accurately represents the Type II error for a two-tailed test when the alternative hypothesis is μ5\mu \neq 5.

Would you like a further breakdown of how β\beta is calculated or more examples on Type I and Type II errors?

Here are some follow-up questions to deepen understanding:

  1. What is the difference between Type I and Type II errors?
  2. How do you interpret the significance level (α\alpha) in hypothesis testing?
  3. Why is it essential to distinguish between one-tailed and two-tailed tests?
  4. How does the sample size nn affect the Type II error (β\beta)?
  5. How would you calculate power (1β1 - \beta) in this scenario?

Tip: In hypothesis testing, carefully distinguish between one-tailed and two-tailed tests to choose the appropriate critical values for α\alpha and β\beta.

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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)