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 will use the formula:
Confidence Interval = (x̄d - tα/2 * (sd/√n), x̄d + tα/2 * (sd/√n))
Where:
- x̄d is the sample mean of the differences
- tα/2 is the critical value from the t-distribution based on the desired confidence level (98% in this case)
- sd is the standard deviation of the sample differences
- n is the sample size
Let's calculate the confidence interval step by step:
1. Calculate the sample mean of the differences (x̄d):
- Add up the differences for each worker: 23-21, 26-24, 19-23, 24-25, 26-24, 22-25, 20-24, 18-23
- Calculate the mean of these differences: (23-21 + 26-24 + 19-23 + 24-25 + 26-24 + 22-25 + 20-24 + 18-23) / 8 = 0.625
Therefore, x̄d = 0.625
2. Calculate the standard deviation of the differences (sd):
- Subtract x̄d from each difference and square the result for each worker: (23-21-0.625)², (26-24-0.625)², (19-23-0.625)², (24-25-0.625)², (26-24-0.625)², (22-25-0.625)², (20-24-0.625)², (18-23-0.625)²
- Sum up these squared differences: (23-21-0.625)² + (26-24-0.625)² + (19-23-0.625)² + (24-25-0.625)² + (26-24-0.625)² + (22-25-0.625)² + (20-24-0.625)² + (18-23-0.625)² = 16.5
To calculate sd, divide the sum by (n-1) and take the square root: sqrt(16.5/7) ≈ 1.6633
Therefore, sd ≈ 1.6633
3. Determine the critical value (tα/2):
- Since we want a 98% confidence interval, the alpha level (α) is 1 - 0.98 = 0.02
- Find the critical value tα/2 for a sample size of 8 and alpha level of 0.02 in the t-distribution table or using statistical software. In this case, tα/2 is approximately 2.821
4. Calculate the standard error (SE):
- Divide sd by the square root of the sample size: 1.6633 / sqrt(8) ≈ 0.5888
Therefore, SE ≈ 0.5888
5. Calculate the lower and upper bounds of the confidence interval:
- Lower bound = x̄d - tα/2 * SE
= 0.625 - 2.821 * 0.5888 ≈ -2.2067
- Upper bound = x̄d + tα/2 * SE
= 0.625 + 2.821 * 0.5888 ≈ 3.4567
6. Write the 98% confidence interval:
The 98% confidence interval for the mean μd of the population paired differences is approximately (-2.2067, 3.4567).
Therefore, we can say with 98% confidence that the true mean difference in assembly time between Machine I and Machine II falls within this interval.