Section 7.6 Assessing Normality 331 105 tickets, what is the probability that everyone who appears for the flight will get a seat? 0.9744 Tech: 0.9745 24. College admissions: A small college has enough space to enroll 300 new students in its incoming freshman class. From past experience, the admissions office knows that 65% of students who are accepted actually enroll. If the admissions office accepts 450 students, what is the probability that there will be enough space for all the students who enroll? 0.7852 Tech: 0.7854 Extending the Concepts 25. Probability of a single number: A fair coin is tossed 100 times. Use the normal approximation to approximate the probability that the coin comes up heads exactly 50 times. 0.0796 Tech: 0.0797 Answers to Check Your Understanding Exercises for Section 7.5 1. No, np = 7.5 < 10. 2. No, n(1 − p) = 7.2 < 10. 3. 0.0721 Tech: 0.0723 4. 0.1251 Tech: 0.1257 SECTION 7.6 Assessing Normality Objectives 1. Use dotplots to assess normality 2. Use boxplots to assess normality 3. Use histograms to assess normality 4. Use stem-and-leaf plots to assess normality 5. Use normal quantile plots to assess normality Many statistical procedures, some of which we will learn about in Chapters 8 and 9, require that we draw a sample from a population whose distribution is approximately normal. Often we don’t know whether the population is approximately normal when we draw the sample. So the only way we have to assess whether the population is approximately normal is to examine the sample. In this section, we will describe some ways in which this can be done. There are three important ideas to remember when assessing normality: 1. We are not trying to determine whether the population is exactly normal. No population encountered in practice is exactly normal. We are only trying to determine whether the population is approximately normal. 2. Assessing normality is more important for small samples than for large samples. When the sample size is large, say n > 30, the Central Limit Theorem ensures that ̄x is approximately normal. Most statistical procedures designed for large samples rely on the Central Limit Theorem for their validity, so normality of the population is not so important in these cases. 3. Hard-and-fast rules do not work well. They are generally too lenient for very small samples (finding populations to be approximately normal when they are not) or too strict for larger samples (finding populations not to be approximately normal when they are). Informal judgment works as well as or better than hard-and-fast rules. Recall: An outlier is a data value that is considerably larger or smaller than most of the rest of the data. When a sample is very small, it is often impossible to be sure whether it came from an approximately normal population. The best we can do is to examine the sample for signs of nonnormality. If no such signs exist, we will treat the population as approximately normal. Because the normal curve is unimodal and symmetric, samples from normal populations rarely have more than one distinct mode, and rarely exhibit a large degree of skewness. In addition, samples from normal populations rarely contain outliers. We summarize the conditions under which we will reject the assumption that a population is approximately normal.
navidi_monk_elementary_statistics_2e_ch7-9
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