Its 13 on the bottom. Continued on the next page. **Section 6.5: The Central Limit Theorem****(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) Interpretation** 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?---**Vital Statistics: Heights of Men**The heights of 18-year-old men are approximately *normally distributed*, with mean 68 inches and standard deviation 3 inches (based on information from *Statistical Abstract of the United States*, 112th edition).**(a)** What is the probability that an 18-year-old man selected at random is between 67 and 69 inches tall?**(b)** If a random sample of nine 18-year-old men is selected, what is the probability that the mean height \( \bar{x} \) is between 67 and 69 inches?**(c) Interpretation** Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?---**Medical: Blood Glucose**Let \( x \) be a random variable that represents the level of glucose in the blood (milligrams per deciliter of blood) after a 12-hour fast. Assume that for people under 50 years old, \( x \) has a distribution that is approximately normal, with mean \( \mu = 85 \) and estimated standard deviation \( \sigma = 25 \) (based on information from *Diagnostic Tests with Nursing Applications*, edited by S. Loeb, Springhouse). A test result \( x < 40 \) is an indication of severe excess insulin, and medication is usually prescribed.**(a)** What is the probability that, on a single test, \( x < 40 \)?**(b)** Suppose a doctor uses the average \( \bar{x} \) for two tests taken about a week apart. What can we say about the probability distribution of \( \bar{x} \)? *Hint*: See Theorem ### Understanding Sampling Distributions**6. Basic Computation: Central Limit Theorem**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 \( \overline{x} \) distribution and compute the mean and standard deviation of the distribution.- **(b)** Find the \( z \) value corresponding to \( \overline{x} = 19 \).- **(c)** Find \( P(\overline{x} < 19) \).- **(d) Interpretation** Would it be unusual for a random sample of size 36 from the \( x \) distribution to have a sample mean less than 19? Explain.**7. Statistical Literacy**- **(a)** If we have a distribution of \( x \) values that is more or less mound-shaped and somewhat symmetric, what is the sample size needed to claim that the distribution of sample means \( \overline{x} \) 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 \( \overline{x} \) taken from random samples of a given size is normal?**8. Critical Thinking**Suppose \( x \) has a distribution with \( \mu = 72 \) and \( \sigma = 8 \).- **(a)** If random samples of size \( n = 16 \) are selected, can we say anything about the \( \overline{x} \) distribution of sample means?- **(b)** If the original \( x \) distribution is normal, can we say anything about the \( \overline{x} \) distribution of random samples of size 16? Find \( P(68 \leq \overline{x} \leq 73) \).**9. Critical Thinking**Consider two \( \overline{x} \) distributions corresponding to the same \( x \) distribution. The first \( \overline{x} \) distribution is based on samples of size \( n = 100 \) and the second is based on samples of size \( n = 225 \). Which \( \overline{x} \) distribution has

Name: Its 13 on the bottom. Continued on the next page. **Section 6.5: The C
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Description: Its 13 on the bottom. Continued on the next page. **Section 6.5: The Central Limit Theorem****(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) Interpretation**

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13. Coal: Automatic Loader 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 u = 75 tons 0.8 ton. and standard deviation o = (a) What is the probability that one car chosen at random will have less than 74.5 tons of coal?