Brainliest + 20 pts to whoever helps pls!! Eighty-one random people were surveyed about the time it takes to commute to work in

Question

Brainliest + 20 pts to whoever helps pls!!

Eighty-one random people were surveyed about the time it takes to commute to work in the morning. The standard deviation of the simple random sample is 2.3 minutes. The test statistic for the sample is 105.8. Find the critical values, using a significance level of 0.10, needed to test a claim that the standard deviation of all commute times is equal to 2.0 minutes. State the initial conclusion.
a. 60.391 and 101.879; because the test statistic is in a critical region, the test rejects the null hypothesis.
b. 60.391 and 101.879; because the test statistic is outside of the critical region, the test fails to reject the null hypothesis.
c. 69.126 and 113.145; because the test statistic is in a critical region, the test rejects the null hypothesis.
d. 69.126 and 113.145; because the test statistic is in the critical region, the test fails to reject the null hypothesis.

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Hailey 3 weeks 2021-11-08T23:26:48+00:00 1 Answer 0 views 0

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    2021-11-08T23:28:46+00:00

    Answer:

    Correct option is (a). 60.391 and 101.879; because the test statistic is in a critical region, the test rejects the null hypothesis.

    Step-by-step explanation:

    A Chi-square test for population variance is used to perform this test.

    The standard deviation is, 2.0 minutes.

    Then the variance is, 4.0 minutes.

    The hypothesis for this test is:

    H₀: The population variance of all commute times is equal to 4.0 minutes, i.e. σ² = 4.

    Hₐ: The population variance of all commute times is not equal to 4.0 minutes, i.e. σ² ≠ 4.

    The test statistic is:

    \chi^{2}_{calc.}=\frac{(n-1)s^{2}}{\sigma^{2}}

    The critical region of this test is defined as:

    Reject H₀ if \chi^{2}_{calc.}<\chi^{2}_{\alpha /2, (n-1)} or \chi^{2}_{calc.}>\chi^{2}_{(1-\alpha /2), (n-1)}.

    The degrees of freedom is:

    n-1=81-1=80

    Compute the critical from a Chi-square table.

    \chi^{2}_{\alpha /2, (n-1)}=\chi^{2}_{0.05, 80}=101.879\\\chi^{2}_{(1-\alpha /2), (n-1)}=\chi^{2}_{0.95, 80}=60.391\\

    The test statistic value is, \chi^{2}_{calc.}=105.8.

    \chi^{2}_{calc.}=105.8 > \chi^{2}_{0.05, 80}=101.879

    The null hypothesis is rejected because the test statistic is in the critical region.

    Thus, the correct option is (a).

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