266 CHAPTER 5 Normal Probability Distributions Probability and the Central Limit Theorem In Section 5.2, you learned how to find the probability that a random variable x will lie in a given interval of population values. In a similar manner, you can find the probability that a sample mean x will lie in a given interval of the x sampling distribution. To transform x to a z@score, you can use the formula z = Value - Mean Standard error = x - mx sx = x - m s 2n . Finding Probabilities for Sampling Distributions The figure at the right shows the mean distances traveled by drivers each day. You randomly select 50 drivers ages 16 to 19. What is the probability that the mean distance traveled each day is between 19.4 and 22.5 miles? Assume s = 6.5 miles. SOLUTION The sample size is greater than 30, so you can use the Central Limit Theorem to conclude that the distribution of sample means is approximately normal, with a mean and a standard deviation of mx = m = 20.7 miles and sx = s1 n = 6.52 50 ≈ 0.9 mile. The graph of this distribution is shown at the left with a shaded area between 19.4 and 22.5 miles. The z@scores that correspond to sample means of 19.4 and 22.5 miles are found as shown. z1 = 19.4 - 20.7 6.5 250 ≈ -1.41 Convert 19.4 to z@score z2 = 22.5 - 20.7 6.5 250 ≈ 1.96 Convert 22.5 to z@score So, using the Standard Normal Table, the probability that the mean distance driven each day by the sample of 50 people is between 19.4 and 22.5 miles is P119.4 6 x 6 22.52 = P1-1.41 6 z 6 1.962 = P1z 6 1.962 - P1z 6 -1.412 = 0.9750 - 0.0793 = 0.8957. Interpretation Of all samples of 50 drivers ages 16 to 19, about 90% will drive a mean distance each day between 19.4 and 22.5 miles, as shown in the graph at the left. This implies that about 10% of such sample means will lie outside the given interval. 30-49 20-29 30.4 30.4 20.7 miles Miles to go The average miles driven each day, by age group: Source: American Automobile Association 37.0 31.0 65-74 50-64 16-19 EXAMPLE 4 μ= 20.7 19.4 22.5 x Mean distance (in miles) Distribution of Sample Means with n = 50 18 19 20 21 22 23 24 1.96 −1.41 0 z z-Score Distribution of Sample Means with n = 50
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