248 CHAPTER 5 Normal Probability Distributions Another way to find normal probabilities is to use technology. You can find normal probabilities using Minitab, Excel, StatCrunch, and the TI-84 Plus. Using Technology to Find Normal Probabilities In the United States, the numbers of physicians involved in patient care per state are normally distributed, with a mean of 280 physicians per 100,000 resident population and a standard deviation of 78 physicians per 100,000 resident population. You randomly select a state. What is the probability that the state has fewer than 300 physicians per 100,000 resident population? Use technology to find the probability. (Adapted from National Center for Health Statistics) SOLUTION Minitab, Excel, StatCrunch, and the TI-84 Plus each have features that allow you to find normal probabilities without first converting to standard z@scores. Note that to use these features, you must specify the mean and standard deviation of the population, as well as any x@values that determine the interval. You are given that m = 280 and s = 78, and you want to find the probability that the state has fewer than 300 physicians per 100,000 resident population, or P1x 6 3002. MINITAB Cumulative Distribution Function Normal with mean = 280 and standard deviation = 78 x P(X … x) 300 0.601183 EXCEL 0.601182965 =NORM.DIST(300,280,78,TRUE) A 1 TI-84 PLUS normalcdf(-10000, 300, 280, 78) .6011829115 STATCRUNCH Normal Calculator Mean: 280 Std. Dev.: 78 P(x … 300) = 0.60118297 From the displays, you can see that P1x 6 3002 ≈ 0.601. Interpretation The probability that the state has fewer than 300 physicians per 100,000 resident population is about 0.601, or 60.1%. TRY IT YOURSELF 3 You randomly select a state. Using the data from Example 3, what is the probability that the state has between 300 and 350 physicians per 100,000 resident population? Use technology to find the probability. Answer: Page A39 EXAMPLE 3 Picturing the World In baseball, a batting average is the number of hits divided by the number of at bats.The batting averages of all Major League Baseball players in a recent year can be approximated by a normal distribution, as shown in the figure. The mean of the batting averages is 0.245 and the standard deviation is 0.017. (Source: Major League Baseball) Batting average μ = 0.245 Major League Baseball 0.22 0.23 0.24 0.25 0.26 0.27 What percent of the players have a batting average of 0.260 or greater? Out of 40 players on a roster, how many would you expect to have a batting average of 0.260 or greater?
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