Question

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.

Answer 1

The critical values are 101.879 and 60.391.

Explanation:

A hypothesis test for variance is performed to test whether the variance of all commute times is equal to 4.0 minutes.

The hypothesis for this test is:

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

Hₐ: The variance of all commute times is equal to 4.0 minutes, i.e.σ² ≠ 4.

The test statistic is:

`\chi^(2)=(ns^(2))/(\sigma^(2))`

Decision rule:

If the test statistic value is less than
`\chi^(2)_((\alpha/2 ),(n-1))` or greater than
`\chi^(2)_((1-\alpha/2 ),(n-1))` then the null hypothesis is rejected.

The significance level of the test is, α = 0.10.

The degrees of freedom is, n - 1 = 81 - 1 = 80

Compute the critical values as follows:

`\chi^(2)_((\alpha/2 ),(n-1))=\chi^(2)_((0.10/2 ),(81-1))=\chi^(2)_(0.05, 80)=101.879`

`\chi^(2)_((1-\alpha/2 ),(n-1))=\chi^(2)_((1-0.10/2 ),(81-1))=\chi^(2)_((0.95),(80))=60.391`

*Use a Chi-square table.

Thus, the critical values are 101.879 and 60.391.