CHAPTER 6 Chapter Quick Quiz 303 Because of our ability to use technology to find exact values of binomial probabilities, the use of a normal approximation to a binomial distribution has become largely obsolete, so this section is included on the website www.TriolaStats.com. Here are two key points: ■ Given probabilities p and q (where q = 1 - p) and sample size n, if the conditions np Ú 5 and nq Ú 5 are both satisfied, then probabilities from a binomial probability distribution can be approximated reasonably well by using a normal distribution having these parameters: m = np s = 2npq ■ The binomial probability distribution is discrete (with whole numbers for the random variable x), but the normal approximation is continuous. To compensate, we use a “continuity correction” with each whole number x represented by the interval from x - 0.5 to x + 0.5. 6-6 Normal as Approximation to Binomial (available at www.TriolaStats.com) Bone Density Test. In Exercises 1–4, assume that scores on a bone mineral density test are normally distributed with a mean of 0 and a standard deviation of 1. 1.Bone Density Sketch a graph showing the shape of the distribution of bone density test scores. 2.Bone Density Find the bone density score that is the 90th percentile, which is the score separating the lowest 90% from the top 10%. 3.Bone Density For a randomly selected subject, find the probability of a bone density score greater than 1.55. 4.Bone Density For a randomly selected subject, find the probability of a bone density score between -1.00 and 2.00. 5. Notation a. Identify the values of m and s for the standard normal distribution. b. What do the symbols mx and sx represent? 6. Salaries It is known that salaries of college professors have a distribution that is skewed. If we repeat the process of randomly selecting 50 college professors and find the mean of each sample, what is the distribution of these sample means? Seat Designs. In Exercises 7–9, assume that when seated, adult males have back-to-knee lengths that are normally distributed with a mean of 23.5 in. and a standard deviation of 1.1 in. (based on anthropometric survey data from Gordon, Churchill, et al.). These data are used often in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats. 7. Find the probability that a male has a back-to-knee length greater than 25.0 in. 8. Find the probability that a male has a back-to-knee length between 22.0 in. and 26.0 in. Chapter Quick Quiz
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