To construct a 95% confidence interval for the difference between the mean delivery times of the two courier services, we can use the formula:
Confidence Interval = (sample mean difference) ± (critical value) * (standard error of the difference)
First, let's calculate the standard error of the difference:
SE = sqrt[(σ1^2 / n1) + (σ2^2 / n2)]
Where:
σ1 = population standard deviation for Courier Service A = 14 minutes
n1 = number of times Courier Service A was used = 60
σ2 = population standard deviation for Courier Service B = 10 minutes
n2 = number of times Courier Service B was used = 48
SE = sqrt[(14^2 / 60) + (10^2 / 48)]
SE = sqrt[(196/60) + (100/48)]
SE = sqrt[3.27 + 2.08]
SE = sqrt(5.35)
SE = 2.31 (rounded to two decimal places)
Next, let's calculate the critical value. Since we want a 95% confidence interval, we need the critical value for a 2-tailed test with alpha level 0.05/2 = 0.025. Looking up this value in the t-distribution table with (n1 + n2 - 2) degrees of freedom (60 + 48 - 2 = 106), we find the critical value to be approximately 1.984.
Now we can substitute the values into the confidence interval formula:
Confidence Interval = (sample mean difference) ± (critical value) * (standard error of the difference)
Confidence Interval = (42 - 38) ± 1.984 * 2.31
Confidence Interval = 4 ± 1.984 * 2.31
Confidence Interval = 4 ± 4.58
Confidence Interval = (-0.58, 8.58)
The lower limit of the 95% confidence interval for the difference between the mean delivery times of the two courier services is -0.58 minutes.
A client used two courier services on a random daily basis over a period of 3 months for similar destinations. Delivery times where noted by a clerk. During this time Courier Service A was used 60 times with a sample average delivery time calculated as 42 minutes. During this same time Courier Service B was used 48 times and a sample average delivery time is calculated as 38 minutes. Suppose that the population standard deviation for Courier Service A is 14 minutes and for Courier Service B is 10 minutes. Construct a 95% confidence interval for the difference between the mean delivery times of the two courier services. Give the lower limit.
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