Question

The mean for the math component of the New Century Achievement Test is reported as 100, with a standard deviation (σ) of 15. A sample of 400 students throughout a particular school district reveals a mean (Symbol) score of 110. Estimate the mean score for all the students in the district, using a 99% confidence interval.

Answer 1

Answer: To estimate the mean score for all the students in the district, we can use a confidence interval. We are given that the sample size is 400, the sample mean is 110, the population means is 100, and the population standard deviation is 15. We are asked to find a 99% confidence interval for the population mean.

The formula for a confidence interval for the population mean with known population standard deviation is:

Confidence interval = sample mean ± z*(σ/√n)

where:

sample mean = 110 (given)

σ = 15 (given)

n = 400 (given)

z = the critical value from the standard normal distribution for a 99% confidence level, which is 2.576.

Substituting the values, we get:

Confidence interval = 110 ± 2.576*(15/√400)

Confidence interval = 110 ± 1.92

Therefore, the 99% confidence interval for the population mean score is (110 - 1.92, 110 + 1.92), or (108.08, 111.92). We can be 99% confident that the true population mean score for all the students in the district falls between 108.08 and 111.92.

Explanation: