Math Problem Statement

Suppose there is a claim that a certain population has a mean greater than 7. Perform a hypothesis test at the 0.10 significance level using a test statistic of 0.871 and a p-value of 0.192. Decide whether to reject the null hypothesis.

Solution

From the image provided, we are working on a hypothesis test with the following details:

Hypotheses:

  • Null hypothesis H0:μ=7H_0: \mu = 7
  • Alternative hypothesis H1:μ>7H_1: \mu > 7

Information provided:

  • Test statistic z=0.871z = 0.871 (rounded to 3 decimal places)
  • pp-value = 0.192 (rounded to 3 decimal places)
  • Level of significance α=0.10\alpha = 0.10

Steps to complete the hypothesis test:

  1. Determine one-tailed or two-tailed test:

    • Since H1:μ>7H_1: \mu > 7, this is a one-tailed test.
  2. Input the test statistic:

    • Test statistic z=0.871z = 0.871.
  3. Shade the area representing the pp-value:

    • For z=0.871z = 0.871, the area to the right represents the pp-value (since this is a one-tailed test).
  4. Input the pp-value:

    • p=0.192p = 0.192.

Decision Rule:

  • If pαp \leq \alpha, reject H0H_0.
  • If p>αp > \alpha, fail to reject H0H_0.

Here, p=0.192p = 0.192 is greater than α=0.10\alpha = 0.10. Hence, we fail to reject H0H_0. This means there is not enough evidence to support the claim that μ>7\mu > 7 at the 0.10 level of significance.


Would you like further explanation or assistance with any specific step?

5 Questions for Expansion:

  1. What is the relationship between pp-value and significance level α\alpha?
  2. How is the test statistic zz calculated in hypothesis testing?
  3. What are the assumptions required for conducting a zz-test for the mean?
  4. How would the decision change if α\alpha were 0.05 instead of 0.10?
  5. Can you explain why this test is one-tailed rather than two-tailed?

Tip:

Always visualize the problem on a normal distribution curve to better understand the area represented by the pp-value, especially for one-tailed and two-tailed tests.

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Math Problem Analysis

Mathematical Concepts

Hypothesis Testing
One-Tailed Test
Significance Level
p-value

Formulas

Test statistic (z) = (sample mean - population mean) / (standard error)
Decision Rule: Reject H0 if p-value <= alpha

Theorems

Central Limit Theorem: Ensures the sampling distribution of the mean is approximately normal.
p-value Method for Hypothesis Testing

Suitable Grade Level

Grades 11-12 or College Level (Statistics)