Question

One significant consequence of the COVID-19 situation is that many meetings that formerly were held face-to-face are now held virtually using something like Zoom or Skype. Suppose the first population is all face-to-face meetings held in March 2020, the second population is all Zoom meetings held in March 2020, and the parameter of interest is μ₁ – μ₂ = the difference in the mean length of all face-to-face meetings and the mean length of all Zoom meetings. The meeting lengths are measured in minutes. For both face-to-face meetings and Zoom meetings the distributions of meeting times are skewed heavily to the right due to some meetings that are very long. A simple random sample of 47 face-to-face meetings held in March 2020 was selected, and the mean length of this sample of 47 meetings was 51 minutes with a standard deviation of 15.6 minutes. An independent simple random sample of 41 Zoom meetings held in March 2020 was selected, and the mean length of this sample of 41 meetings was 54 minutes with a standard deviation of 14.0 minutes. If appropriate, use this information to calculate and interpret a 99% confidence interval for the difference in the mean length of all face-to-face meetings and the mean length of all Zoom meetings.

Answer 1

Solution:

•99% CI for mean difference is (-11.375, 5.375)

Calculation:

We need to construct the 99% confidence interval for the difference between the population means Mi - M2, for the case that the population standard deviations are not known. The following information has been provided about each of the samples: Sample Mean 1 (X1) = 51 Sample Standard Deviation 1 (81) = 15.6 Sample Size 1 (N1) = 47 Sample Mean 2 (#2) = 54 Sample Standard Deviation 2 (82) = 14 Sample Size 2 (N2) = Based on the information provided, we assume that the population variances are equal, so then the number of degrees of freedom are df = ni + n2 – 2 = 47 + 41 – 2 = 86. The critical value for a = 0.01 and df = 86 degrees of freedom is te = ti-a/2:n-1 = 2.634. The corresponding confidence interval is computed as shown below: Since the population variances are assumed to be equal, we need to compute the pooled standard deviation, as follows: Sp = (nı - 1)sí + (n2 – 1)s nu + n2 – 2 (47 – 1) 15.62 + (41 – 1) × 142 = VS 47 + 41 – 2 = 14.877 Since we assume that the population variances are equal, the standard error is computed as follows: se = SpVn + "V ni se = saying that = 14.877 * Vantaa = 3.179 Now, we finally compute the confidence interval: CI = (X1 - X2 – te X se, ł1 - 72 +te x se) = (51 – 54 – 2.634 x 3.179,51 – 54 + 2.634 x 3.179) = (-11.375,5.375) Therefore, based on the data provided, the 99% confidence interval for the difference between the population means ji – M2 is -11.375 < Mi - M2 < 5.375, which indicates that we are 99% confident that the true difference between population means is contained by the interval (-11.375,5.375).