The human resources department of a consulting firm gives a standard creativity test to a randomly selected group of new hires every year. This year, 50 new hires took the test and scored a mean of 111.6 points with a standard deviation of 14.3. Last year, 55 new hires took the test and scored a mean of 118.4 points with a standard deviation of 15.6 . Assume that the population standard deviations of the test scores of all new hires in the current year and the test scores of all new hires last year can be estimated by the sample standard deviations, as the samples used were quite large. Construct a 99% confidence interval for u1-u2 , the difference between the mean test score u1 of new hires from the current year and the mean test score of u2 new hires from last year.
To construct the confidence interval for the difference between the mean test scores of new hires from the current year and last year, we can use the formula:
CI = (x1 - x2) ± Z * √((s1^2 / n1) + (s2^2 / n2))
Where:
- CI is the confidence interval
- x1 and x2 are the sample means
- s1 and s2 are the sample standard deviations
- n1 and n2 are the sample sizes
- Z is the critical value based on the desired confidence level
Given:
- For the current year: x1 = 111.6, s1 = 14.3, n1 = 50
- For last year: x2 = 118.4, s2 = 15.6, n2 = 55
- Confidence level: 99%
Step 1: Calculate the critical value (Z)
The critical value represents the number of standard deviations needed to obtain a specific level of confidence. For a 99% confidence level, we need to find the Z-value that leaves only 1% of the distribution in the tails.
Using a Z-table or calculator, we find that the Z-value for a 99% confidence level is approximately 2.576.
Step 2: Calculate the standard error (SE)
The standard error measures the variability of the sample means. It is calculated using the formula:
SE = √((s1^2 / n1) + (s2^2 / n2))
Plugging in the values, we get:
SE = √((14.3^2 / 50) + (15.6^2 / 55))
Step 3: Calculate the margin of error (ME)
The margin of error represents the range in which the true difference between means lies. It is calculated by multiplying the critical value (Z) by the standard error (SE).
ME = Z * SE
Plugging in the values, we get:
ME = 2.576 * SE
Step 4: Calculate the confidence interval (CI)
Finally, we can construct the confidence interval by subtracting and adding the margin of error (ME) to the difference between the sample means (x1 - x2).
CI = (x1 - x2) ± ME
Plugging in the values, we get:
CI = (111.6 - 118.4) ± ME
Step 5: Calculate the confidence interval
Calculate the mean difference between the two sample means (x1 - x2):
Mean Difference = 111.6 - 118.4
Now, plug in the calculated values to get the final confidence interval.