The variabilities of the service times of two employees at the office of Registrations and records...

The variabilities of the service times of two employees at the office of Registrations and records at UW-Stout are of interest. The supervisor wants to determine whether the variance of service time for the 1^st employee is greater than t6hat for the 2^nd employee.

Data

Employees	Random Sample	Mean Times	Standard Deviations
1	8	3.4 minutes	1.8 minutes
2	10	2.5 minutes	0.9 minutes

Find the confidence interval for the ratio of the two variances (Use alpha = 0.05)
Can we conclude that the variance of the service time is greater for the 1^st employee than the 2^nd?

Solutions

Expert Solution

With two tail F test, you just want to know if the variances are not equal to each other. In notation:
H_a = σ²1 ≠ σ² 2

Sample 1: Variance = (1.8)^ = 3.24, sample size = 8
Sample 2: Variance = (0.9)^2 = 0.81, sample size = 10

hypothesis statements:
H_o: No difference in variances.
H_a: Difference in variances.

F Statistic = variance 1/ variance 2 = 3.24 / 0.81

= 4

Critical F (7,9) at alpha (0.05) = 4.197

a) confidence interval at 0.05 =

The degrees of freedom will be the sample size -1, so:
Sample 1 has 7 df (the numerator).
Sample 2 has 9 df (the denominator).

F_0.025 = 4.197

F_0.975 = 0.207

(3.24/0.81)/4.197 < σ₁²/ σ₂² < (3.24/.81)/0.207

0.953 < σ₁²/ σ₂² < 19.323

Since from the sample information we get that FL =0.207 < F=4 < FU =4.197, it is then concluded that the null hypothesis is not rejected.

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that employee 1 is greater than the employee 2 at 0.05α=0.05significance level.

orchestra answered 1 year ago

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