Suppose a new standardized test is given to 100 randomly selected third-grade students in New Jersey. The sample average score, \(\bar{Y}\), on the test is 58 points, and the sample standard deviation, \(s_{Y}\), is 8 points.

a. The authors plan to administer the test to all third-grade students in New Jersey. Construct a 95\% confidence interval for the mean score of all New Jersey third graders.

b. Suppose the same test is given to 200 randomly selected third graders from Iowa, producing a sample average of 62 points and a sample standard deviation of 11 points. Construct a \(90 \%\) confidence interval for the difference in mean scores between Iowa and New Jersey.

c. Can you conclude with a high degree of confidence that the population means for Iowa and New Jersey students are different? (What is the standard error of the difference in the two sample means? What is the \(p\)-value of the test of no difference in means versus some difference?)