Textbook Exercises (a) Population = all four-year U.S. college students, Sample = 17,096 selected students

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3 Textbook Exercises 3.84 (a) Population = all undergrad college students ages 18 to 24 (b) Sample = 2036 undergrad college students who participate in the survey 3.91 (a) Population = all four-year U.S. college students, Sample = 17,096 selected students 3.98 (b) Population = all restaurant workers, Sample = 100 selected restaurant workers (c) Population = 584 longleaf pines in the tract, Sample = 40 trees that were measured The larger sample size (n = 100 vs. n = 25) will reduce sampling variability so that sample statistics are likely to be closer to population parameters (a) P (x 21) = P (Z 0.41) = (b) µ x = µ x = 18.6 σ x = σx n = = (c) P ( x 21) = P (Z 2.88) = = (a) P ( x > 2) = P (Z > 4.71) 0 (b) The actual sample mean is x = 520/200 = 2.6 flaws per square yard Since x = 2.6 > 2, this is an almost impossible sample outcome according to (a), assuming that the claim that µ = 1.6 flaws is true. Therefore the claim must be false! (This is called a logical proof by contradiction.) But if µ = 1.6 is false, what s the true value of µ? Our best estimate for µ is x = 2.6 from the sample, so it appears that the original claim undercounts the mean number of flaws in the carpet material. (Notice also that a carpet manufacturer interested in carpet sales has an incentive to overstate the claimed quality of the carpet.) P ( x > 15%) = P ( x < 7%) = The number 19 is a parameter and 14 is a statistic These numbers are parameters since they re calculated from census information. 3

4 4.117 (a) Since individual measurements x are normally distributed, x is also normally distributed for any sample size n, as illustrated by the graph in Topic 6 Example 2. (A large sample size n 30 isn t required.) The mean and standard deviation of the bell curve for x are µ x = µ x = 123 σ x = σx n = = (b) P ( x 124) = P (Z 21.65) 0 (off the table) µ x = mm, σ x = mm P ( x > ) = (a) P (x > 60) = (b) P (x < 60) = 1 P (x > 60) = (c) P ( x > 60) = (a) If only 12 policies are sold (with total revenue $ = $3000), it would be a financial disaster for the company if even one home burns down! But selling thousands of policies generates a lot more revenue to protect the company. (b) P ( x > 275) = Between and average defects (a) The variable x = number of people in a car is actually a discrete variable since the possible counts for x are 0, 1, 2,.... But a normally-distributed variable is continuous, as the Topic 6 Notes explain. So the variable x in this exercise isn t normally-distributed. (b) The sample size is large (n = 700 > 30) so the sample mean x has an approximate normal distribution with mean and standard deviation (c) µ x = 1.5 persons/car σ x = persons/car 4

5 5.77 Chapter 5 Exercises success: answer question correctly p = 0.75 x = number of questions answered correctly on the test (a) If n = 100, σ p = P ( p 0.70) = P (Z 1.15) = (b) If n = 250, P ( p 0.70) = P (Z 1.83) = What are English definitions for success and the variable x in this exercise? (a) µ = np = 704(0.328) = (b) P (x 262) = home runs (c) The chances for an average baseball player are approximately 0. Did this feat require a monumental effort by Suzuki? Probably so, since his chance in any year is less than 1% success: order ships within three days p = 0.90 n = 100 x = number of orders from the sample which are shipped within three days (a) σ p = 0.03 P (x 86) = (b) Even if the 90% claim is true, we expect some variation from sample to sample. In fact, there is about a 9% chance for 86 or fewer items from 100 to be shipped within three days. This probability is not so small as to definitely disprove the claim. success: survey contacts black American adult p = 0.12 n = 1500 x = number of black American adults contacted in the survey (a) 180 persons and persons, respectively (b) The answer is So if the survey plan is representative of black Americans, there is still about a 21% chance that any particular sample of 1500 will include 170 or fewer blacks The flaw is that a bell curve for the sample proportion p does not apply since the sample size is too small: n = 12 < 30. (See Some Final Notes About Topic 6 on Notebook page 246.) 5