Math Problem Statement

Test the claim about the population variance sigma squared at the level of significance alpha. Assume the population is normally distributed. ​Claim: sigma squaredless than or equals18.2​; alphaequals0.10 Sample​ statistics: s squaredequals26.52​, nequals41 Question content area bottom Part 1 Write the null and alternative hypotheses. Upper H 0​: sigma squared ▼ greater than less than or equals equals not equals less than greater than or equals    enter your response here Upper H Subscript a​: sigma squared ▼ less than or equals not equals equals less than greater than or equals greater than    enter your response here

Solution

To test the claim about the population variance (σ2\sigma^2) at the level of significance α=0.10\alpha = 0.10, we need to set up the null and alternative hypotheses.

Given:

  • The claim is that the population variance is less than or equal to 18.2 (σ218.2\sigma^2 \leq 18.2).
  • The sample variance is s2=26.52s^2 = 26.52.
  • The sample size is n=41n = 41.
  • The significance level α=0.10\alpha = 0.10.

Hypotheses:

  • Null Hypothesis (H0H_0): This is the statement we are trying to test, and it usually reflects the idea that there is no effect or no difference. The claim that the variance is less than or equal to 18.2 is a statement of equality, so the null hypothesis is:

    H0:σ218.2H_0: \sigma^2 \leq 18.2

  • Alternative Hypothesis (HaH_a): This is the opposite of the null hypothesis, reflecting what we want to test. Since the claim is that the variance is less than or equal to 18.2, the alternative hypothesis would state that the population variance is greater than 18.2. So, the alternative hypothesis is:

    Ha:σ2>18.2H_a: \sigma^2 > 18.2

Thus, the hypotheses are:

  • H0:σ218.2H_0: \sigma^2 \leq 18.2
  • Ha:σ2>18.2H_a: \sigma^2 > 18.2

Next Steps:

  • You can now calculate the test statistic and compare it to the critical value (using the Chi-Square distribution) to make a decision.
  • Do you need help calculating the test statistic and making a decision?

Further Questions:

  1. What is the test statistic for this hypothesis test?
  2. How do you find the critical value for this Chi-Square test?
  3. What is the decision rule for this hypothesis test?
  4. How do you interpret the results in the context of the claim?
  5. How does the significance level (α\alpha) affect the hypothesis testing?

Tip: Always double-check the direction of the inequality in your hypotheses to ensure they align with the claim and the test you're performing!

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Math Problem Analysis

Mathematical Concepts

Hypothesis Testing
Chi-Square Distribution
Population Variance

Formulas

Chi-Square test statistic formula: χ² = (n-1) * s² / σ₀²

Theorems

Chi-Square Test for Variance

Suitable Grade Level

Grades 11-12