Question

Statistics Grades: The statistics grades in the fall semester had mean of 65. The SRS were taken from different statistics groups (A, B, C and D) considering that the total number of students that took part on the exam was 110. Assuming that the change in the grades has a normal distribution with standard deviation σ= 10, We computed a 90% confidence interval of the mean change in score μ in the population of all statistics students.

A) Find a 90% confidence interval for μ based on this sample.

B) What is the margin of error of 90%? How does decreasing the confidence level (example 80%) change the margin of error of a confidence interval when the sample size and population standard deviation remain the same? Calculate the interval for a confidence level of 80% What is the meaning of reduced margin of error?

C) Suppose we had an SRS of just 80 students. What would be the margin of error for 90% confidence interval?

D) How does decreasing the sample size change the margin of error of a confidence interval when the confidence level and population standard deviation remain the same? Use a) and c) in order to explain your answer.

E) Because of different size samples, draw two different graphs with the confidence level and intervals. In the first graph draw the results of a), b) and in the second graph draw the result of c) and d). Mark each interval the confidence level. Explain if the density curve differs when the sample size changes.

Answer 1

A) Since population SD is known, z score would be used

So required 90% CI for u,

65 +- 1.645*10/√110

= (63.432, 66.568)

B) margin of error = 1.645*10/√110 = 1.568

As confidence level decreases to 80% margin of error decreases, since z score for 80% confidence is given as 1.28

So the margin of error becomes, 1.28*10/√110 = 1.220

So the confidence interval becomes,

65 +- 1.220

=(63.78, 66.22)

C) at n=80, moe= 1.645*10/√80 = 1.839

D) decreasing the sample size from 110 to 80 has increased the margin of error when other things remain the same (from 1.568 to 1.839)