Final answer:
To construct a 90% confidence interval for the population mean of errors per day by bank tellers, we use the t-distribution due to the small sample size. The confidence interval is calculated by adding and subtracting the margin of error, derived from the critical t-value and standard error, to the sample mean.
Step-by-step explanation:
The subject of the question is to construct a 90% confidence interval for the population mean number of errors bank tellers make per day. Given the sample mean is 3.6 and the sample standard deviation is 0.42 with a sample size of 12, we can calculate the confidence interval using the t-distribution because the sample size is small.
First, find the critical t-value for a 90% confidence level and 11 degrees of freedom (n-1). Next, calculate the standard error by dividing the sample standard deviation by the square root of the sample size. Then, multiply the standard error with the critical t-value to find the margin of error. Finally, add and subtract the margin of error from the sample mean to get the confidence interval.
The step-by-step calculation would look like this:
- Find the critical t-value for 11 degrees of freedom.
- Calculate the standard error (SE = s / √n, where s is the sample standard deviation and n is the sample size).
- Multiply the SE by the critical t-value to find the margin of error.
- Add and subtract the margin of error from the sample mean to find the confidence interval.
For instance, if the t-value for a 90% confidence level and 11 degrees of freedom is approximately 1.8, the margin of error would be 1.8*(0.42/√12), and the confidence interval would be 3.6 ± that margin of error, leading to one of the available options in the question.