SECTION 5.4 Sampling Distributions and the Central Limit Theorem 267 TRY IT YOURSELF 4 You randomly select 100 drivers ages 30 to 49 from Example 4. What is the probability that the mean distance traveled each day is between 36.2 and 37.8 miles? Assume s = 5.8 miles. Answer: Page A39 Finding Probabilities for Sampling Distributions In a recent year, the mean room and board expense at four-year colleges was $11,806. You randomly select 9 four-year colleges. What is the probability that the mean room and board was less than $12,250? Assume that the room and board expenses are normally distributed with a standard deviation of $1650. (Adapted from National Center for Education Statistics) SOLUTION Because the population is normally distributed, you can use the Central Limit Theorem to conclude that the distribution of sample means is normally distributed, with a mean and a standard deviation of mx = m = $11,806 and sx = s1 n = $1650 2 9 = $550. The graph of this distribution is shown at the left. The area to the left of $12,250 is shaded. The z@score that corresponds to $12,250 is z = 12,250 - 11,806 1650 29 = 444 550 ≈ 0.81. So, using the Standard Normal Table, the probability that the mean room and board expense was less than $12,250 is P1x 6 12,2502 = P1z 6 0.812 = 0.7910. You can check this answer using technology. For instance, you can use a TI-84 Plus to find the x-value, as shown below. TI-84 PLUS normalcdf(-10000,12250, 11806,550) 0.7902453767 Interpretation So, about 79% of such samples with n = 9 will have a mean less than $12,250 and about 21% of these sample means will be greater than $12,250. TRY IT YOURSELF 5 In a recent year, the mean existing home sales price in the United States was $296,700. You randomly select 12 existing homes. What is the probability that the mean sales price was more than $275,000? Assume that the sales prices are normally distributed with a standard deviation of $50,000. (Adapted from National Association of Realtors) Answer: Page A39 EXAMPLE 5 Study Tip Before you find probabilities for intervals of the sample mean x, use the Central Limit Theorem to determine the mean of the sample means mx and the standard deviation of the sample means sx. μ= 11,806 12,250 x Mean room and board (in dollars) Distribution of Sample Means with n = 9 7000 9000 11,000 13,000 15,000 17,000
RkJQdWJsaXNoZXIy NjM5ODQ=