In a time-use study, 20 randomly selected managers were found to spend a mean of 2.40 hours each day on paperwork. The standard deviation of the the 20 scores is 1.30. Also, the sample data appear to have a bell-shaped distribution. Construct the 95% confidence interval for the mean time spent on paperwork by all managers.
To construct a confidence interval for the mean time spent on paperwork by all managers, you can use the following formula:
Confidence interval = sample mean ± (critical value) * (standard deviation / sqrt(sample size))
In this case, the sample mean is 2.40 hours, the standard deviation is 1.30, and the sample size is 20 managers.
First, you need to find the critical value for a 95% confidence level. Since the sample data appear to have a bell-shaped distribution, you can assume a normal distribution and use the Z-table.
To find the critical value, you need to determine the z-score corresponding to a 95% confidence level. The z-score for a 95% confidence level is approximately 1.96. This can be obtained from the z-table or using a calculator.
Once you have the critical value, you can plug in the values into the formula:
Confidence interval = 2.40 ± (1.96) * (1.30 / sqrt(20))
Simplifying the calculation:
Confidence interval = 2.40 ± (1.96) * (1.30 / 4.47)
Confidence interval = 2.40 ± (1.96) * (0.29)
Confidence interval = 2.40 ± 0.57
So, the 95% confidence interval for the mean time spent on paperwork by all managers is (1.83, 2.97) hours.
To construct a 95% confidence interval for the mean time spent on paperwork by all managers, we can use the following formula:
Confidence Interval = Sample Mean ± (Critical Value) * (Standard Deviation / √n)
In this case:
- The sample mean is 2.40 hours.
- The standard deviation is 1.30.
- The sample size is 20.
- The critical value for a 95% confidence interval is 1.96.
Substituting these values into the formula, we get:
Confidence Interval = 2.40 ± (1.96) * (1.30 / √20)
Simplifying the expression:
Confidence Interval = 2.40 ± (1.96) * (0.29)
Calculating the values:
Confidence Interval = 2.40 ± 0.567
Therefore, the 95% confidence interval for the mean time spent on paperwork by all managers is:
Lower Limit = 2.40 - 0.567 = 1.833
Upper Limit = 2.40 + 0.567 = 2.967
So, the 95% confidence interval for the mean time spent on paperwork by all managers is approximately 1.833 to 2.967 hours.
Formula:
CI95 = mean ± 1.96(sd/√n)
Your data:
mean = 2.40
sd = 1.30
n = 20
Plug the values into the formula and calculate the interval.
I hope this will help get you started.