268 CHAPTER 5 Normal Probability Distributions The Central Limit Theorem can also be used to investigate unusual events. An unusual event is one that occurs with a probability of less than 5%. Finding Probabilities for x and x Some college students use credit cards to pay for school-related expenses. For this population, the amount paid is normally distributed, with a mean of $2172 and a standard deviation of $740. (Adapted from Sallie Mae/Ipsos) 1. What is the probability that a randomly selected college student, who uses a credit card to pay for school-related expenses, paid less than $1900? 2. You randomly select 25 college students who use credit cards to pay for school-related expenses. What is the probability that their mean amount paid is less than $1900? 3. Compare the probabilities from parts 1 and 2. SOLUTION 1. In this case, you are asked to find the probability associated with a certain value of the random variable x. The z@score that corresponds to x = $1900 is z = x - m s = 1900 - 2172 740 = -272 740 ≈ -0.37. So, the probability that the student paid less than $1900 is P1x 6 19002 = P1z 6 -0.372 = 0.3557. You can check this answer using technology. For instance, you can use Excel to find the probability, as shown at the left. (The answer differs slightly due to rounding.) 2. Here, you are asked to find the probability associated with a sample mean x. The z@score that corresponds to x = $1900 is z = x - mx sx = x - m s 2n = 1900 - 2172 740 225 = -272 148 ≈ -1.84. So, the probability that the mean credit card balance of the 25 card holders is less than $1900 is P1x 6 19002 = P1z 6 -1.842 = 0.0329. You can check this answer using technology. For instance, you can use Excel to find the probability, as shown at the left. (The answer differs slightly due to rounding.) 3. Interpretation Although there is about a 36% chance that a college student who uses a credit card to pay for school-related expenses will pay less than $1900, there is only about a 3% chance that the mean amount a sample of 25 college students will pay is less than $1900. Because there is only a 3% chance that the mean amount a sample of 25 college students will pay is less than $1900, this is an unusual event. TRY IT YOURSELF 6 A consumer price analyst claims that prices for computer monitors are normally distributed, with a mean of $208 and a standard deviation of $107. What is the probability that a randomly selected monitor costs less than $200? You randomly select 10 monitors. What is the probability that their mean cost is less than $200? Compare these two probabilities. Answer: Page A39 EXAMPLE 6 Study Tip To find probabilities for individual members of a population with a normally distributed random variable x, use the formula z = x - m s . To find probabilities for the mean x of a sample of size n, use the formula z = x - mx sx . EXCEL 0.356597851 =NORM.DIST(1900,2172,740,TRUE) A 1 EXCEL 0.033043152 =NORM.DIST(1900,2172,148,TRUE) A 1
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