 ## (Solution document) Suppose that the number of pairs of clothes in a year by Americans is normally distributed and the mean number of clothes is 20 a year.

1. Suppose that the number of pairs of clothes in a year by Americans is normally distributed and the mean number of clothes is 20 a year. Suppose that a researcher takes 1,990,000 samples of Americans, and each sample consists of 5 Americans. She asks each American how many pairs of clothes they buy in a year.  She then calculates a mean for each of the 1,990,000 samples. If she added up the mean for each sample, and divided by 1,990,000, what would she get? Explain why.

5 points

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2. Not only is the mean of pairs of clothes 20, but the population standard deviation is 1.8. When the researcher from problem number 1 calculates a standard deviation for the distribution of sample means for her 1,990,000 samples, is the standard deviation of sample means likely to be less than, greater than, or equal to 1.8? Explain.                                                                5 points

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3. A widely used measure of health of infants is the body weight. Since body weight has been used so frequently, we have a pretty good idea of the population mean and standard deviation for scores on the body weight. The mean body weight is 10, and the standard deviation is 2. Given this, what is the probability that:                                                                                                    2 points each

a) In a sample of 100 infants, the body weight is 12 pounds or greater.

b) In a sample of 100 infants, the body weight is 12 or less.

c) In a sample of 36 infants, the body weight is 12 or less.

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4. Explain why the probability for getting a mean body weight of 12 or less in Question 3 differs between parts b and c.                                                                                                   4 points