The purchasing department has suggested that all new computer monitors for your company should
be a new design. You want data to assure you that employees will like the new monitors. The next 20
employees needing a new computer are the subjects for an experiment.
(a) Label the employees 01 to 20. Randomly choose 10 to receive new monitors. The
remaining 10 get standard monitors.
(b) After a month of use, employees express their satisfaction with their new monitors by
responding to the statement "I like my new monitor" on a scale from 1 to 5, where 1 represents
"strongly disagree", 2 is "disagree", 3 is "neutral", 4 is "agree", and 5 stands for "strongly agree'.
The employees with the new monitors have average satisfaction 4.4 with standard deviation 0.7.
The employees with the standard monitors have average 3.2 with standard deviation 1.5. Give a
95% con�dence interval for the di�erence in the mean satisfaction scores for all employees.
To calculate the 95% confidence interval for the difference in mean satisfaction scores for all employees, you can follow these steps:
Step 1: Calculate the mean and standard deviation for the group with the new monitors:
- The average satisfaction for employees with new monitors is 4.4.
- The standard deviation of satisfaction for employees with new monitors is 0.7.
Step 2: Calculate the mean and standard deviation for the group with standard monitors:
- The average satisfaction for employees with standard monitors is 3.2.
- The standard deviation of satisfaction for employees with standard monitors is 1.5.
Step 3: Determine the sample sizes of the two groups:
- The number of employees with new monitors is 10.
- The number of employees with standard monitors is also 10.
Step 4: Calculate the pooled standard deviation (Sp) using the formula:
Sp = sqrt(((n1-1)*s1^2 + (n2-1)*s2^2) / (n1 + n2 - 2))
- where n1 and n2 are the sample sizes and s1 and s2 are the standard deviations of the two groups.
Plugging in the values:
Sp = sqrt(((10-1)*(0.7^2) + (10-1)*(1.5^2)) / (10 + 10 - 2))
= sqrt((9*0.49 + 9*2.25) / 18)
= sqrt((4.41 + 20.25) / 18)
= sqrt(24.66 / 18)
≈ sqrt(1.37)
≈ 1.17
Step 5: Calculate the standard error (SE) using the formula:
SE = Sp * sqrt(1/n1 + 1/n2)
Plugging in the values:
SE = 1.17 * sqrt(1/10 + 1/10)
= 1.17 * sqrt(0.2 + 0.2)
= 1.17 * sqrt(0.4)
≈ 1.17 * 0.632
≈ 0.741
Step 6: Calculate the margin of error (ME) using the formula:
ME = t * SE
- where t is the critical t-value with (n1 + n2 - 2) degrees of freedom. For a 95% confidence level and 18 degrees of freedom, the t-value can be determined from a t-table or calculator. Let's assume the t-value is 2.101 (approximately).
Plugging in the values:
ME = 2.101 * 0.741
≈ 1.556
Step 7: Calculate the confidence interval:
Confidence Interval = (mean of new monitors - mean of standard monitors) ± ME
Plugging in the values:
Confidence Interval = (4.4 - 3.2) ± 1.556
= 1.2 ± 1.556
= (approximately) -0.356 to 2.756
Therefore, the 95% confidence interval for the difference in mean satisfaction scores for all employees is approximately -0.356 to 2.756.