310 Chapter 7 The Normal Distribution d. Between what two values are the middle 60% of the scores? 68.28 and 81.72 Tech: 68.27 and 81.73 Extending the Concepts 51. Tall men: Heights of men in a certain city are normally distributed with mean 70 inches. Sixteen percent of the men are more than 73 inches tall. What percentage of the men are between 67 and 70 inches tall? 34% 52. Watch your speed: Speeds of automobiles on a certain stretch of freeway at 11:00 P.M. are normally distributed with mean 65 mph. Twenty percent of the cars are traveling at speeds between 55 and 65 mph. What percentage of the cars are going faster than 75 mph? 30% 53. Contaminated wells: A study reported that the mean concentration of ammonium in water wells in the state of Iowa was 0.71 milligram per liter, and the standard deviation was 1.09 milligrams per liter. Is it possible to determine whether these concentrations are approximately normally distributed? If so, say whether they are normally distributed, and explain how you know. If not, describe the additional information you would need to determine whether they are normally distributed. Not normal Source: Water Environment Research 74:177–186 54. Heights: According to the National Health Statistics Reports, heights of adult women in the United States are normally distributed with mean 64 inches and standard deviation 4 inches. If three women are selected at random, what is the probability that at least one of them is more than 68 inches tall? 0.5955 Tech: 0.5956 55. Exam scores: Scores on an exam were normally distributed. Ten percent of the scores were below 64 and 80% were below 81. Find the mean and standard deviation of the scores. Mean: 74.26, SD: 8.02 Tech: 8.01 56. Commute to work: Megan drives to work each morning. Her commute time is normally distributed with mean 30 minutes and standard deviation 4 minutes. Her workday begins at 9:00 A.M. At what time should she leave for work so that the probability she is on time is 95%? 8:23 A.M. Answers to Check Your Understanding Exercises for Section 7.2 1. z = −1. Interpretation: A value of 11 is one standard deviation below the mean. 2. z = 0.75. Interpretation: A value of 75 is 0.75 standard deviation above the mean. 3. z = −0.4. Interpretation: The length of time for this bulb is 0.4 standard deviation below the mean. 4. 0.0228 5. 0.0968 6. 0.6898 7. 9.084 Tech: 9.088 8. 66.26 Tech: 66.10 SECTION 7.3 Sampling Distributions and the Central Limit Theorem Objectives 1. Construct the sampling distribution of a sample mean 2. Use the Central Limit Theorem to compute probabilities involving sample means In Section 7.2, we learned to compute probabilities for a randomly sampled individual from a normal population. In practice, statistical studies involve sampling several, perhaps many, individuals. As discussed in Chapter 3, we often compute numerical summaries of samples, and the most commonly used summary is the sample mean ̄x. If several samples are drawn from a population, they are likely to have different values for ̄x. Because the value of ̄x varies each time a sample is drawn, ̄x is a random variable, and it has a probability distribution. The probability distribution of ̄x is called the sampling distribution of ̄x. Objective 1 Construct the sampling distribution of a sample mean An Example of a Sampling Distribution Tetrahedral dice are four-sided dice, used in role-playing games such as Dungeons & Dragons. They are shaped like a pyramid, with four triangular faces. Each face corresponds to a number between 1 and 4, so that when you toss a tetrahedral die, it comes up with one of the numbers 1, 2, 3, or 4. Tossing a tetrahedral die is like sampling a value from the population 1 2 3 4 The population mean, variance, and standard deviation are: Population mean: �� = 1 + 2 + 3 + 4 4 = 2.5 Population variance: ��2 = (1 − 2.5)2 + (2 − 2.5)2 + (3 − 2.5)2 + (4 − 2.5)2 4 = 1.25
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