Question

. (0.5 point) We simulate the operations of a call center that opens from 8am to 6pm for 20 days. The daily average call waiting times are reported below for these 20 simulated days: 24.16 20.17 14.60 19.79 20.02 14.60 21.84 21.45 16.23 19.60 17.64 16.53 17.93 22.81 18.05 16.36 15.16 19.24 18.84 20.77 Due to a budget cut, the call center has now to downsize and the simulation model is used to predict the effect. We again simulate the call center for 20 days and now the daily average call waiting times are: 19.81 18.39 24.34 22.63 20.20 23.35 16.21 21.73 17.18 18.98 19.35 18.41 20.57 13.00 17.25 21.32 23.29 22.09 12.88 19.27 Compute a 95% t-confidence interval to decide if the change in the call waiting time is statistically significant.

Answer 1

The 95% t-confidence interval for the difference in mean is approximately (-2.61, 1.16), therefore, there is not enough statistical evidence to show that there is a change in waiting time, therefore;

The change in the call waiting time is not statistically significant

Explanation:

The given call waiting times are;

24.16, 20.17, 14.60, 19.79, 20.02, 14.60, 21.84, 21.45, 16.23, 19.60, 17.64, 16.53, 17.93, 22.81, 18.05, 16.36, 15.16, 19.24, 18.84, 20.77

19.81, 18.39, 24.34, 22.63, 20.20, 23.35, 16.21, 21.73, 17.18, 18.98, 19.35, 18.41, 20.57, 13.00, 17.25, 21.32, 23.29, 22.09, 12.88, 19.27

From the data we have;

The mean waiting time before the downsize,
`\overline x_1` = 18.7895

The mean waiting time before the downsize, s₁ = 2.705152

The sample size for the before the downsize, n₁ = 20

The mean waiting time after the downsize,
`\overline x_2` = 19.5125

The mean waiting time after the downsize, s₂ = 3.155945

The sample size for the after the downsize, n₂ = 20

The degrees of freedom, df = n₁ + n₂ - 2 = 20 + 20 - 2 = 38

df = 38

At 95% significance level, using a graphing calculator, we have;
`t_(\alpha /2)` = ±2.026192

The t-confidence interval is given as follows;

`\left (\bar{x}_(1)- \bar{x}_(2) \right )\pm t_(\alpha /2)\sqrt{(s_(1)^(2))/(n_(1))+(s_(2)^(2))/(n_(2))}`

Therefore;

`\left (18.7895- 19.5152 \right )\pm 2.026192 * \sqrt{(2.705152^(2))/(20)+(3.155945^2)/(20)}`

(18.7895 - 19.5125) - 2.026192*(2.705152²/20 + 3.155945²/20)^(0.5)

The 95% CI = -2.6063 < μ₂ - μ₁ < 1.16025996668

By approximation, we have;

The 95% CI = -2.61 < μ₂ - μ₁ < 1.16

Given that the 95% confidence interval ranges from a positive to a negative value, we are 95% sure that the confidence interval includes '0', therefore, there is sufficient evidence that there is no difference between the two means, and the change in call waiting time is not statistically significant.