Final answer:
The 90% confidence interval for the mean grading time of all composition papers, based on a sample of 40 English composition professors with a sample mean of 12 minutes and a population standard deviation of 4 minutes, is approximately 10.96 to 13.04 minutes.
Step-by-step explanation:
The question is about finding the 90% confidence interval for the mean grading time of all composition papers based on a sample of 40 English composition professors. Given is the sample mean (μ) as 12 minutes and the population standard deviation (σ) as 4 minutes. Utilizing the formula for a confidence interval and the Z-score associated with a 90% confidence level, we can calculate the range in which we are 90% confident the true mean grading time lies.
To compute the confidence interval, the following formula is used:
Confidence Interval = μ ± (Z*[σ/√n])
where μ is the sample mean, Z is the Z-score for the desired confidence level, σ is the population standard deviation, and n is the sample size.
For a 90% confidence level, the Z-score is approximately 1.645. Plugging in the given values into the formula:
Confidence Interval = 12 ± (1.645*[4/√40])
Which simplifies to:
Confidence Interval = 12 ± (1.645*[4/6.3246]) ≈ 12 ± 1.037
Therefore, the 90% confidence interval for the mean grading time is approximately 10.96 to 13.04 minutes. We can say with 90% confidence that the true mean grading time for all composition papers falls within this interval.