Question

The filling variance for boxes of cereal is designed to be .02 or less. A sample of 41 boxes of cereal shows a sample standard deviation of .16 ounces.

Use α = .05 to determine whether the variance in the cereal box fillings is exceeding the design specification.

Answer 1

`\chi^2 =(41-1)/(0.02) 0.0256 =51.2`

`p_v =P(\chi^2_(40) >51.2)=0.1104`

In order to find the p value we can use the following code in excel:

"=1-CHISQ.DIST(51.2,40,TRUE)"

If we compare the p value and the significance level provided 0.05 we see that
`p_v >\alpha` so on this case we have enough evidence in order to FAIL reject the null hypothesis at the significance level provided. And that means that the population variance is not significantly higher than 0.02, so there is no a violation of the specifications.

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=41` represent the sample size

`\alpha=0.05` represent the confidence level

`s^2 =0.16^2 =0.0256` represent the sample variance obtained

`\sigma^2_0 =0.02` represent the value that we want to test

Null and alternative hypothesis

On this case we want to check if the population variance specification is higher than the standard of 0.02, so the system of hypothesis would be:

Null Hypothesis:
`\sigma^2 \leq 0.02`

Alternative hypothesis:
`\sigma^2 >0.02`

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 =(41-1)/(0.02) 0.0256 =51.2`

Calculate the p value

In order to calculate the p value we need to have in count the degrees of freedom , on this case
`df = 41-1=40`. And since is a right tailed test the p value would be given by:

`p_v =P(\chi^2_(40) >51.2)=0.1104`

In order to find the p value we can use the following code in excel:

"=1-CHISQ.DIST(51.2,40,TRUE)"

Conclusion

If we compare the p value and the significance level provided 0.05 we see that
`p_v >\alpha` so on this case we have enough evidence in order to FAIL reject the null hypothesis at the significance level provided. And that means that the population variance is not significantly higher than 0.02, so there is no a violation of the specifications.