A comparison of the last year's (Population 1) versus the current year's (Population 2) shipments (in thousands of dollars) for each of six Canadian exporters is given as follows: (4.34, 4.77), (5.89, 5.16), (2.52, 2.73), (3.91, 4.39), (4.99, 5.25), (3.24, 3.80). It is assumed that the population of differences between the last year’s and the current year's shipments is approximately normal. Then a 95% confidence interval estimate for the mean difference is:
Question 9 options:
-0.5787 to 0.1753
-0.6765 to 0.2731
-0.5926 to 0.1893
-0.7005 to 0.2972
To calculate the confidence interval estimate for the mean difference between the last year's and the current year's shipments, we can use the following steps:
1. Calculate the differences between the last year's and the current year's shipments for each Canadian exporter. In this case, the differences are:
- 4.77 - 4.34 = 0.43
- 5.16 - 5.89 = -0.73
- 2.73 - 2.52 = 0.21
- 4.39 - 3.91 = 0.48
- 5.25 - 4.99 = 0.26
- 3.80 - 3.24 = 0.56
2. Calculate the sample mean of the differences. Add up all the differences and divide by the number of differences. In this case, the sum of the differences is 0.43 + (-0.73) + 0.21 + 0.48 + 0.26 + 0.56 = 1.31, and there are 6 differences, so the sample mean is 1.31 / 6 = 0.2183.
3. Calculate the sample standard deviation of the differences. This can be done by finding the squared differences, summing them up, dividing by the number of differences minus 1, and then taking the square root. In this case, the squared differences are:
- (0.43 - 0.2183)^2 = 0.04896989
- (-0.73 - 0.2183)^2 = 0.80083089
- (0.21 - 0.2183)^2 = 0.00067489
- (0.48 - 0.2183)^2 = 0.07024689
- (0.26 - 0.2183)^2 = 0.00176729
- (0.56 - 0.2183)^2 = 0.11613689
The sum of the squared differences is 0.04896989 + 0.80083089 + 0.00067489 + 0.07024689 + 0.00176729 + 0.11613689 = 1.03862664. Dividing this by the number of differences minus 1 (6-1 = 5) gives us the sample variance: 1.03862664 / 5 = 0.207725328, and taking the square root gives us the sample standard deviation: sqrt(0.207725328) = 0.455204516.
4. Calculate the standard error of the mean difference. Divide the sample standard deviation by the square root of the number of differences. In this case, the standard error is 0.455204516 / sqrt(6) = 0.185948313.
5. Determine the appropriate critical value from the t-distribution based on the desired confidence level. Since the confidence level is 95%, we want to find the critical value for a 2-tailed test at alpha = 0.05. With 5 degrees of freedom (6-1 = 5) and a 95% confidence level, the critical value from the t-distribution is approximately 2.571.
6. Calculate the margin of error. Multiply the standard error of the mean difference by the critical value. In this case, the margin of error is 0.185948313 * 2.571 = 0.478568574.
7. Calculate the lower and upper bounds of the confidence interval. Subtract the margin of error from the sample mean difference to get the lower bound, and add the margin of error to the sample mean difference to get the upper bound. In this case, the lower bound is 0.2183 - 0.478568574 = -0.260268574, and the upper bound is 0.2183 + 0.478568574 = 0.696868574.
Therefore, the 95% confidence interval estimate for the mean difference is approximately -0.26 to 0.697.
None of the options listed match exactly with this confidence interval estimate.