Question

n = 23 H0: σ2 ≥ 66 s2 = 60 Ha: σ2 < 66 The test statistic has a value of _____. a. 24.20 b. 24.00 c. 20.91 d. 20.00

Answer 1

`\chi^2 =(23-1)/(66) 60 =20.00`

d.20.00

Explanation:

Notation and previous concepts

A chi-square test is "used to test if the variance of a population is equal to a specified value. This test can be either a two-sided test or a one-sided test. The two-sided version tests against the alternative that the true variance is either less than or greater than the specified value"

`n=23` represent the sample size

`\alpha` represent the confidence level

`s^2 =60` represent the sample variance obtained

`\sigma^2_0 =66` represent the value that we want to test

Null and alternative hypothesis

On this case we want to check if the population variance is less than 66, so the system of hypothesis would be:

Null Hypothesis:
`\sigma^2 \geq 66`

Alternative hypothesis:
`\sigma^2 <66`

Calculate the statistic

For this test we can use the following statistic:

`\chi^2 =(n-1)/(\sigma^2_0) s^2`

And this statistic is distributed chi square with n-1 degrees of freedom. We have eveything to replace.

`\chi^2 =(23-1)/(66) 60 =20.00`

Calculate the p value

In order to calculate the p value we need to have in count the degrees of freedom , on this case 23-1=22. And since is a left tailed test the p value would be given by:

`p_v =P(\chi^2 <20.00)=0.417`

The best option is:

d. 20.00