Question

The grades of a sample of 10 applicants, selected at random from a large population, are 71, 86, 75, 63, 92, 70, 81, 59, 80, and 90. Compute the sample variance. Can we infer at the 90% confidence that the population variance is significantly less than 100? (i.e. Perform Hypothesis testing)

Answer 1

Solution:

Here, we have to find the value for sample variance.

From given data, we have sample variance = S^2 = 123.1223

(by using excel)

Now, we have to use the Chi square test for population variance. The null and alternative hypotheses for this test are given as below:

Null hypothesis: H₀: The population variance is 100.

Alternative hypothesis: H_a: The population variance is significantly less than 100.

H₀: σ² = 100 versus H_a: σ² < 100

This is lower tailed test or left tailed test.

We are given

Confidence level = c = 90% = 0.90

So, significance level = α = 1 – c = 1 – 0.90 = 0.10 or 10%

The test statistic formula is given as below:

Chi square = (n – 1)*S^2/ σ²

From given data, we have

n = 10

df = n – 1 = 10 – 1 = 9

S^2 = 123.1223

σ² = 100

Chi square = (10 - 1)* 123.1223/100

Chi square = 11.08101

P-value = 0.7298

P-value > α = 0.10

So, we do not reject the null hypothesis

There is insufficient evidence to conclude that the population variance is significantly less than 100.