Question

3. A small business has 37 employees. Because of the uncertain demand for its product, the

company usually pays overtime on any given week. The manager assumed that about 50

hours of overtime per week is required and that the variance on this figure is about 25.

Company officials want to know whether the variance of overtime hours has changed in the

last year. The last 16 weeks of overtime data yielded a sample variance of 22. Does this data

support a conclusion at the 0.05 significance level that the variance of overtime hours has

decreased?

4. Develop a 95% confidence interval estimate of the variance in overtime hours in the previous

problem.

Answer 1

3)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: σ^2 = 25
Alternative Hypothesis, Ha: σ^2 < 25

Rejection Region
This is left tailed test, for α = 0.05 and df = 15
Critical value of Χ^2 is 7.261
Hence reject H0 if Χ^2 < 7.261

Test statistic,
Χ^2 = (n-1)*s^2/σ^2
Χ^2 = (16 - 1)*22/25
Χ^2 = 13.2

P-value Approach
P-value = 0.4131
As P-value >= 0.05, fail to reject null hypothesis.

4)

Here s = 4.6904 and n = 16
df = 16 - 1 = 15

α = 1 - 0.95 = 0.05
The critical values for α = 0.05 and df = 15 are Χ^2(1-α/2,n-1) = 6.262 and Χ^2(α/2,n-1) = 27.488

CI = (15*4.6904^2/27.488 , 15*4.6904^2/6.262)
CI = (12.0052 , 52.6985)