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# NAEP scores Young people have a better chance of full-time employment and good wages if they aregood with numbers. How strong are the quantitative skills of young Americans of working age? One source of data is the National Assessment of Educational Progress (NAEP) Young Adult Literacy Assessment Survey, which is based on a nationwide probability sample of households. The NAEP survey includes a short test of quantitative skills, covering mainly basic arithmetic and the ability to apply it to realistic problems. Scores on the test range from 0 to 500. For example, a person who scores 233 can add the amounts of two checks appearing on a bank deposit slip; someone scoring 325 can determine the price of a meal from a menu; a person scoring 375 can transform a price in cents per ounce into dollars per pound. $^{4}$ Suppose that you give the NAEP test to an SRS of840 people from a large population in which the scores have mean 280 and standard deviation S 60. The mean $\overline{x}$ of the 840 scores will vary if you take repeated samples.(a) Describe the shape, center, and spread of the sampling distribution of $\overline{x} .$(b) Sketch the sampling distribution of $\overline{x}$ . Mark its mean and the values one, two, and three standard deviations on either side of the mean.(c) According to the $68-95-99.7$ rule, about 95$\%$ of all values of $\overline{x}$ lie within a distance $m$ of the mean of the sampling distribution. What is $m ?$ Shade the region on the axis of your sketch that is within $m$ of the mean.(d) Whenever $\overline{x}$ falls in the region you shaded, the population mean $\mu$ lies in the confidence interval $\overline{x} \pm m .$ For what percent of all possible samples does the interval capture $\mu$ ?

## a. Approximately normal with mean 280 and standard deviation 2.0702b. see drawingc. $m=4.140$d. 95$\%$

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