A time study of a certain service task found an average time of 15 minutes, with a standard deviation of 5minutes.

These figures were based on a sample of 100 measurements. Is the sample large enough that we are 95% confident that standard time is within 5% of its true value?

To determine if the sample size is large enough to be 95% confident that the standard time is within 5% of its true value, we need to calculate the margin of error and compare it to the desired 5% (0.05).

The margin of error can be calculated using the formula:

Margin of Error = Z * (Standard Deviation / √n)

Where:
- Z is the Z-score corresponding to the desired confidence level (95% confidence corresponds to a Z-score of 1.96).
- Standard Deviation is the population standard deviation, which is estimated using the sample standard deviation (5 minutes in this case).
- n is the sample size (100 measurements in this case).

Plugging in the given values, the margin of error is calculated as:

Margin of Error = 1.96 * (5 / √100) = 1.96 * (5 / 10) = 0.98

Next, we need to determine the range within which the standard time should fall. To find the range, we need to calculate 5% of the expected standard time:

5% of Standard Time = 0.05 * 15 minutes = 0.75 minutes

Finally, we compare the margin of error to the range. If the margin of error is less than the range, then the sample size is large enough to be 95% confident that the standard time is within 5% of its true value.

In this case, the margin of error (0.98 minutes) is greater than the range (0.75 minutes), indicating that the sample size of 100 measurements is large enough to be 95% confident that the standard time is within 5% of its true value.