TESTING TWO POPULATION MEANS AND PROPORTIONS 63 Test Statistic: (We show the full decimal values for the calculations for df and the test statistic in case you want to track the calculations yourself.) Observed Significance Level: To find the p- value for the t test statistic, we use the function in Excel =tdist(2.37275,56,2). p- value = 0.0211 Conclusion: Since the test statistic of t = ­2.373 falls well below the lower critical bound of t = ­2.003, we reject H 0 with at least 95% confidence. Likewise, since the p- value of 0.0211 is less than the desired of 0.05, we reject H 0 . There is enough evidence to conclude that there is a difference in the average wait times for customer service at the two offices. The F- Test for Equality of Two Variances When we sample randomly from the same population, sample vari- ances do vary simply because we randomly select different subsets of the population. But how different do variances have to be before we are concerned that they are not equal? To test the equality of population variances, whether 2 = 2 , we form the ratio of the 1 2 two variances by dividing both sides of the equation by one of the variances. We then determine if the resulting ratio is significantly dif- ferent from 1. The best estimate for related sample statistics is the ratio of the two . When two independent samples are taken from normally distributed populations with equal variances, the sampling distribution of their ratios follows an F-distribution, named after the English statistician Sir Ronald A. Fisher (1890­1962). The F-distribu- tion includes the degrees of freedom for each of the two samples: n 1 ­ 1 degrees of freedom for the numerator and n 2 ­ 1 for the denominator.