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Does a raw score less than the mean correspond to a positive or negative standard score? What about a raw score greater than the mean?

Suppose \(5 \%\) of the area under the standard normal curve lies to the left of \(z\). Is \(z\) positive or negative?

Suppose \(x\) has a distribution with a mean of 20 and a standard deviation of \(3 .\) Random samples of size \(n=36\) are drawn. (a) Describe the \(\bar{x}\) distribution and compute the mean and standard deviation of the distribution. (b) Find the \(z\) value corresponding to \(\bar{x}=19\). (c) Find \(P(\bar{x}<19)\). (d) Would it be unusual for a random sample of size 36 from the \(x\) distribution to have a sample mean less than 19? Explain.

(a) If we have a distribution of \(x\) values that is more or less mound-shaped and somewhat symmetrical, what is the sample size needed to claim that the distribution of sample means \(\bar{x}\) from random samples of that size is approximately normal? (b) If the original distribution of \(x\) values is known to be normal, do we need to make any restriction about sample size in order to claim that the distribution of sample means \(\bar{x}\) taken from random samples of a given size is normal?

Assuming that the heights of college women are normally distributed with mean 65 inches and standard deviation \(2.5\) inches (based on information from Statistical Abstract of the United States, 112 th Edition), answer the following questions. Hint: Use Problems 5 and 6 and Figure \(6-3\). (a) What percentage of women are taller than 65 inches? (b) What percentage of women are shorter than 65 inches? (c) What percentage of women are between \(62.5\) inches and \(67.5\) inches? (d) What percentage of women are between 60 inches and 70 inches?

Do you try to pad an insurance claim to cover your deductible? About \(40 \%\) of all U.S. adults will try to pad their insurance claims! (Source: Are You Normal?, by Bernice Kanner, St. Martin's Press.) Suppose that you are the director of an insurance adjustment office. Your office has just received 128 insurance claims to be processed in the next few days. What is the probability that (a) half or more of the claims have been padded? (b) fewer than 45 of the claims have been padded? (c) from 40 to 64 of the claims have been padded? (d) more than 80 of the claims have not been padded?

The college physical education department offered an advanced first aid course last semester. The scores on the comprehensive final exam were normally distributed, and the \(z\) scores for some of the students are shown below: $$ \begin{array}{lcl} \text { Robert, } 1.10 & \text { Juan, } 1.70 & \text { Susan, }-2.00 \\ \text { Joel, } 0.00 & \text { Jan, }-0.80 & \text { Linda, } 1.60 \end{array} $$ (a) Which of these students scored above the mean? (b) Which of these students scored on the mean? (c) Which of these students scored below the mean? (d) If the mean score was \(\mu=150\) with standard deviation \(\sigma=20\), what was the final exam score for each student?

Consider an \(x\) distribution with standard deviation \(\sigma=12\). (a) If specifications for a research project require the standard error of the corresponding \(\bar{x}\) distribution to be 2, how large does the sample size need to be? (b) If specifications for a research project require the standard error of the corresponding \(\bar{x}\) distribution to be 1, how large does the sample size need to be?

Coal is carried from a mine in West Virginia to a power plant in New York in hopper cars on a long train. The automatic hopper car loader is set to put 75 tons of coal into each car. The actual weights of coal loaded into each car are normally distributed, with mean \(\mu=75\) tons and standard deviation \(\sigma=0.8\) ton. (a) What is the probability that one car chosen at random will have less than \(74.5\) tons of coal? (b) What is the probability that 20 cars chosen at random will have a mean load weight \(\bar{x}\) of less than \(74.5\) tons of coal? (c) Suppose the weight of coal in one car was less than \(74.5\) tons. Would that fact make you suspect that the loader had slipped out of adjustment? Suppose the weight of coal in 20 cars selected at random had an average \(\bar{x}\) of less than \(74.5\) tons. Would that fact make you suspect that the loader had slipped out of adjustment? Why?

Let \(x\) be a random variable that represents the weights in kilograms (kg) of healthy adult female deer (does) in December in Mesa Verde National Park. Then \(x\) has a distribution that is approximately normal, with mean \(\mu=63.0 \mathrm{~kg}\) and standard deviation \(\sigma=7.1 \mathrm{~kg}\) (Source: The Mule Deer of Mesa Verde National Park, by G. W. Mierau and J. L. Schmidt, Mesa Verde Museum Association). Suppose a doe that weighs less than \(54 \mathrm{~kg}\) is considered undernourished. (a) What is the probability that a single doe captured (weighed and released) at random in December is undernourished? (b) If the park has about 2200 does, what number do you expect to be undernourished in December? (c) To estimate the health of the December doe population, park rangers use the rule that the average weight of \(n=50\) does should be more than \(60 \mathrm{~kg}\). If the average weight is less than \(60 \mathrm{~kg}\), it is thought that the entire population of does might be undernourished. What is the probability that the average weight \(\bar{x}\) for a random sample of 50 does is less than \(60 \mathrm{~kg}\) (assume a healthy population)? (d) Compute the probability that \(\bar{x}<64.2 \mathrm{~kg}\) for 50 does (assume a healthy population). Suppose park rangers captured, weighed, and released 50 does in December, and the average weight was \(\bar{x}=64.2 \mathrm{~kg}\). Do you think the doe population is undernourished or not? Explain.