USES AND ABUSES Statistics in the Real World 284 CHAPTER 5 Normal Probability Distributions EXERCISES 1. Is It Unusual? A population is normally distributed, with a mean of 100 and a standard deviation of 15. Determine whether either event is unusual. Explain your reasoning. a. The mean of a sample of 3 is 112 or more. b. The mean of a sample of 75 is 105 or more. 2. Find the Error The mean age of students at a high school is 16.5, with a standard deviation of 0.7. You use the Standard Normal Table to determine that the probability of selecting one student at random whose age is more than 17.5 years is about 8%. What is the error in this problem? 3. Give an example of a distribution that might be nonnormal. Uses Normal distributions can be used to describe many real-life situations and are widely used in the fields of science, business, and psychology. They are the most important probability distributions in statistics and can be used to approximate other distributions, such as discrete binomial distributions. The most incredible application of the normal distribution lies in the Central Limit Theorem. This theorem states that no matter what type of distribution a population may have, as long as the size of each random sample is at least 30, the distribution of sample means will be approximately normal. When a population is normal, the distribution of sample means is normal for any random sample of size n. The normal distribution is essential to sampling theory. Sampling theory forms the basis of statistical inference, which you will study in the next chapter. Abuses Consider a population that is normally distributed, with a mean of 100 and standard deviation of 15. It would not be unusual for an individual value taken from this population to be 115 or more. In fact, this will happen almost 16% of the time. It would be, however, highly unusual to take a random sample of 100 values from that population and obtain a sample mean of 115 or more. Because the population is normally distributed, the mean of the sampling distribution of sample means will be 100, and the standard deviation will be 1.5. A sample mean of 115 lies 10 standard deviations above the mean. This would be an extremely unusual event. When an event this unusual occurs, it is a good idea to question the original parameters or the assumption that the population is normally distributed. Although normal distributions are common in many populations, you should not try to make nonnormal statistics fit a normal distribution. The statistics used for normal distributions are often inappropriate when the distribution is nonnormal. For instance, some economists argue that financial risk managers’ reliance on normal distributions to model stock market behavior is a mistake because the normal distributions do not accurately predict unusual events like market crashes.
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