15. A report states the proportion of employees at a large company who worked overtime in the last month.
A poll randomly surveyed 200 employees, and 12% stated they had worked overtime in the last month.
The company reports with 95% confidence that less than 1 out of 6 employees worked overtime in the
last month. Evaluate the conclusion of this report
To evaluate the conclusion of this report, we need to compare the reported proportion (12%) with the claim made by the company (less than 1 out of 6, or 16.7%).
First, we can calculate the margin of error for the sample proportion using the formula:
margin of error = critical value x standard error
The critical value for a 95% confidence interval with a sample size of 200 is 1.96. The standard error can be calculated as:
standard error = sqrt(p(1-p)/n)
where p is the sample proportion and n is the sample size. Plugging in the values, we get:
standard error = sqrt(0.12(1-0.12)/200) = 0.031
So the margin of error is:
margin of error = 1.96 x 0.031 = 0.061
This means that we can be 95% confident that the true population proportion of employees who worked overtime in the last month is within 6.1 percentage points of the sample proportion (12%).
Next, we can calculate the upper bound of a one-sided 95% confidence interval using the formula:
upper bound = sample proportion + margin of error
upper bound = 0.12 + 0.061 = 0.181
So we can say with 95% confidence that the true population proportion of employees who worked overtime in the last month is less than 18.1%.
Finally, we compare this result to the claim made by the company that less than 1 out of 6 employees worked overtime in the last month. One sixth is equivalent to 16.7%, which is slightly higher than our upper bound of 18.1%. This means that the report's conclusion is consistent with the company's claim, and we can accept the conclusion that less than 1 out of 6 employees worked overtime in the last month with 95% confidence.