Amie Jackson,a manager at Sigma travel services,makes every effort to ensure that customers attempting to make online reservations do not have to wait too long to complete the reservation process.The travel website is open for reservations 24 hours a day,and Amie regularly checks the website for the waiting time to maintain consistency in service.She uses the following independently drawn samples of wait time during two peak hours,morning 8a.m.to 10 a.m.and evening 7p.m.to 9p.m. ,for the analysis.Assume that wait times are normally distributed. Construct a 90% confidence interval and test if the ratio of the population variances differs from one at the 10% significance level.Which of the following outcomes supports the test?
A) [1.47,5.99].The ratio of the population variances differs from one at the 10% significance level.
B) [1.19,7.35].The ratio of the population variances differs from one at the 10% significance level.
C) [1.47,5.99].The ratio of the population variances does not differ from one at the 10% significance level.
D) [1.19,7.35].The ratio of the population variances does not differ from one at the 10% significance level.

Question

Amie Jackson,a manager at Sigma travel services,makes every effort to ensure that customers attempting to make online reservations do not have to wait too long to complete the reservation process.The travel website is open for reservations 24 hours a day,and Amie regularly checks the website for the waiting time to maintain consistency in service.She uses the following independently drawn samples of wait time during two peak hours,morning 8a.m.to 10 a.m.and evening 7p.m.to 9p.m. ,for the analysis.Assume that wait times are normally distributed.  Construct a 90% confidence interval and test if the ratio of the population variances differs from one at the 10% significance level.Which of the following outcomes supports the test?
A) [1.47,5.99].The ratio of the population variances differs from one at the 10% significance level. 
B) [1.19,7.35].The ratio of the population variances differs from one at the 10% significance level. 
C) [1.47,5.99].The ratio of the population variances does not differ from one at the 10% significance level. 
D) [1.19,7.35].The ratio of the population variances does not differ from one at the 10% significance level.

Quizplus · Accepted Answer

To construct a 90% confidence interval for the ratio of population variances, we use the formula:

((n1-1)s1^2)/(n2-1)s2^2) * (F_(α/2))(v2,v1) ≤ σ1^2/σ2^2 ≤ ((n1-1)s1^2)/(n2-1)s2^2) * (F_(1-α/2))(v1,v2)

Where s1^2 and s2^2 are the sample variances, n1 and n2 are the sample sizes, and F_(α/2)(v1,v2) and F_(1-α/2)(v1,v2) are the values of the F-distribution with v1 and v2 degrees of freedom at the α/2 and 1-α/2 percentiles, respectively.

Using the given data, we have:

n1 = 25, s1^2 = 7.4, v1 = n1 - 1 = 24
n2 = 25, s2^2 = 5.7, v2 = n2 - 1 = 24
α = 0.1
F_(α/2)(v1,v2) = F_(0.05)(24,24) ≈ 0.419
F_(1-α/2)(v1,v2) = F_(0.95)(24,24) ≈ 2.840

Substituting these values into the formula, we get:

1.19 ≤ σ1^2/σ2^2 ≤ 7.35

To test if the ratio of population variances differs from one at the 10% significance level, we use the F-test. The null hypothesis is H0: σ1^2/σ2^2 = 1, and the alternative hypothesis is Ha: σ1^2/σ2^2 ≠ 1.

The test statistic is F = s1^2/s2^2. Under the null hypothesis, this follows an F-distribution with v1 and v2 degrees of freedom. The critical values for a two-tailed test at the 10% significance level are F_(0.05)(24,24) ≈ 0.419 and F_(0.95)(24,24) ≈ 2.840.

If the test statistic falls outside this range, we reject the null hypothesis and conclude that the ratio of population variances differs from one. Otherwise, we fail to reject the null hypothesis and conclude that there is not enough evidence to say that the ratio of population variances differs from one.

Using the given data, we have:

F = s1^2/s2^2 = 7.4/5.7 ≈ 1.30

Since 0.419 ≤ F ≤ 2.840, we fail to reject the null hypothesis and conclude that there is not enough evidence to say that the ratio of population variances differs from one at the 10% significance level.

Option B is the only one with the correct 90% confidence interval and conclusion.

Amie Jackson,a Manager at Sigma Travel Services,makes Every Effort to Ensure