A company is considering installing new machines to assemble its products. The company is considering two types of machines, but it will buy only one type. The company selected eight assembly workers and asked them to use these two types of machines to assemble products. The following table gives the time taken (in minutes) to assemble one unit of the product on each type of machine for each of these eight workers.
Machine I 23 26 19 24 26 22 20 18
Machine II 21 24 23 25 24 25 24 23
Construct a 98% confidence interval for the mean μd of the population paired differences, where a paired difference is equal to the time taken to assemble a unit of the product on Machine I minus the time taken to assemble a unit of the product on Machine II.
To construct a confidence interval for the mean μd of the population paired differences, we can follow these steps:
Step 1: Calculate the paired differences for each worker by subtracting the time taken to assemble a unit on Machine II from the time taken on Machine I:
Differences (d):
(23-21) = 2
(26-24) = 2
(19-23) = -4
(24-25) = -1
(26-24) = 2
(22-25) = -3
(20-24) = -4
(18-23) = -5
Step 2: Calculate the sample mean (denoted by x̄d) and sample standard deviation (denoted by sd) of the paired differences:
Sample Mean(x̄d) = (2+2-4-1+2-3-4-5)/8 = -1/8 = -0.125
Sample Standard Deviation (sd) = sqrt((Σd^2 - x̄d^2)/(n-1))
= sqrt((4+4+16+1+4+9+16+25 - (-1/8)^2)/(8-1))
= sqrt((77 + 1/64)/7)
= sqrt((4928 + 1)/448)
= sqrt(69.9888)
= 8.3651
Step 3: Determine the critical value for a 98% confidence interval. Since the sample size is small (n = 8), we will use a t-distribution. With a confidence level of 98% and degrees of freedom (df) = n-1 = 7, the critical value is t = 3.499.
Step 4: Calculate the margin of error (ME) using the formula:
Margin of Error (ME) = t * (sd / sqrt(n))
ME = 3.499 * (8.3651 / sqrt(8)) = 11.5207
Step 5: Calculate the confidence interval by subtracting and adding the margin of error from the sample mean:
Confidence Interval = x̄d ± ME
= -0.125 ± 11.5207
= (-11.6457, 11.3957)
Therefore, the 98% confidence interval for the mean μd of the population paired differences is (-11.6457, 11.3957).