High School

A real estate agent claims that the average price of a home in a certain zip code is less than $250,000. To test this claim, the real estate agent selects 20 homes at random from the zip code and calculates the mean and standard deviation for home prices from the sample. Assume the population of home prices is normally distributed. The dataset for home prices is given below. The setup for the null and alternative hypothesis for this hypothesis test is as follows: H0: μ ≥ 250000; Ha: μ < 250000, which is a left-tailed test. The test statistic and p-value are given as follows: t = -2.082, p-value = 0.026. Which of the following are appropriate conclusions for this hypothesis test? Select all that apply: Reject H0. Fail to reject H0. At the 5% significance level, the data provide sufficient evidence to support the claim that the average price of a home in a certain zip code is less than $250,000. At the 5% significance level, the data do not provide sufficient evidence to support the claim that the average price of a home in a certain zip code is less than $250,000.

Answer :

To determine the appropriate conclusions for this hypothesis test, we need to analyze the given information and compare it to the significance level.

Step-by-Step Explanation:

  1. Hypotheses Setup:

    • The null hypothesis [tex]H_0[/tex]: [tex]\mu \geq 250,000[/tex].
    • The alternative hypothesis [tex]H_a[/tex]: [tex]\mu < 250,000[/tex].
    • This is a left-tailed test since we want to test if the mean is less than $250,000.

  2. Significance Level:

    • The test is conducted at a 5% significance level, or [tex]\alpha = 0.05[/tex].

  3. Test Statistic and p-value:

    • The test statistic is given as [tex]t = -2.082[/tex].
    • The p-value is provided as 0.026.

  4. Decision Rule:

    • Compare the p-value to the significance level [tex]\alpha[/tex]:
      • If [tex]\text{p-value} < \alpha[/tex], reject [tex]H_0[/tex].
      • If [tex]\text{p-value} \geq \alpha[/tex], fail to reject [tex]H_0[/tex].

  5. Comparison and Conclusion:

    • In this case, [tex]0.026 < 0.05[/tex], so we reject [tex]H_0[/tex].
    • At the 5% significance level, the data provide sufficient evidence to support the claim that the average price of a home in this zip code is less than $250,000.

Chosen Multiple Choice Options:

  • Reject H0.
  • At the 5% significance level, the data provide sufficient evidence to support the claim that the average price of a home in a certain zip code is less than $250,000.

These selections are appropriate because the p-value is less than the significance level, leading us to reject the null hypothesis.

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