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16.1 Number Skills of Eighth-Graders. The National Assessment of Educational Progress (NAEP) includes a mathematics test for eighth-grade students. 3 Scores on the test range from 0 to 500 . Demonstrating the ability to use the mean to solve a problem is an example of the skills and knowledge associated with performance at the Basic level. An example of the knowledge and skills associated with the Proficient level is being able to read and interpret a stem-andleaf plot. In \(2019,147,400\) eighth-graders were in the NAEP sample for the mathematics test. The mean mathematics score was \(x=282\). We want to estimate the mean score \(\mu\) in the population of all eighth-graders. Consider the NAEP sample as an SRS from a Normal population with standard deviation \(\sigma=40\) a. If we take many samples, the sample mean \(x\) varies from sample to sample according to a Normal distribution with mean equal to the unknown mean score \(\mu\) in the population. What is the standard deviation of this sampling distribution? b. According to the 95 part of the 68-95-99.7 rule, \(95 \%\) of all values of \(x\) fall within on either side of the unknown mean \(\mu\). What is the missing number? c. What is the \(95 \%\) confidence interval for the population mean score \(\mu\) based on this one sample?

Retaking the SAT. An SRS of 400 high school seniors gained an average of \(x=40\) points in their second attempt at the SAT Mathematics exam. Assume that the change in score has a Normal distribution with standard deviation \(\sigma=25\). We want to estimate the mean change in score \(\mu\) in the population of all high school seniors. a. Give a \(95 \%\) confidence interval for \(\mu\) based on this sample. b. Based on your confidence interval in part (a), how certain are you that the mean change in score \(\mu\) in the population of all high school seniors is greater than 0 ? (Hint: Does the interval in part (a) include 0?)

Retaking the SAT. In Exercise \(16.2\) (page 371), we saw that an SRS of 400 high school seniors gained an average of \(x=40\) points in their second attempt at the SAT Mathematics exam. Assuming that the change in score has a Normal distribution with standard deviation \(\sigma=25\), we computed a \(95 \%\) confidence interval for the mean change in score \(\mu\) in the population of all high school seniors. a. Find a \(90 \%\) confidence interval for \(\mu\) based on this sample. b. What is the margin of error for a confidence level of \(90 \%\) ? How does decreasing the confidence level change the margin of error of a confidence interval when the sample size and population standard deviation remain the same? c. Suppose we had an SRS of just 100 high school seniors. What would be the margin of error for \(95 \%\) confidence? 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?