Question

1. The sample variance of a random sample of 50 observations from a normal population was found to be s² = 80. Can we infer at the 1% significance level (i.e., a = .01) that the population variance is less than 100 (i.e., x < 100) ?
2. Repeat part a changing the sample size to 100
3. What is the affect of increasing the sample size?

Answer 1

Solution:

Here, we have to use Chi square test for the population variance.

H0: σ2 = 100 versus Ha: σ2 < 100

This is a lower tailed test.

The level of significance is given as α = 0.01.

The test statistic formula is given as below:

Chi-square = (n – 1)*S^2/ σ2

Chi-square = (50 – 1)*80/100

Chi-square = 49*80/100

Chi-square =39.2

Degrees of freedom = n – 1 = 50 - 1 = 49

P-value = 0.1596

(by using Chi square table)

P-value > α = 0.01

So, we do not reject the null hypothesis

We cannot infer at the 1% significance level that the population variance is less than 100.

Now, we have to repeat above part by using sample size n as 100.

Chi-square = (n – 1)*S^2/ σ2

Chi-square = (100 – 1)*80/100

Chi-square = 99*80/100

Chi-square =79.2

Degrees of freedom = n – 1 = 100 – 1 = 99

P-value = 0.0714

(by using Chi square table)

P-value > α = 0.01

So, we do not reject the null hypothesis

We cannot infer at the 1% significance level that the population variance is less than 100.

There is no effect of increasing sample size on the final result, but the p-value is decreases as the sample size increases.