Test 2C AP Statistics Name: Part 1: Multiple Choice. Circle the letter corresponding to the best answer. 1. Which of these variables is least likely to have a Normal distribution? (a) Annual income for all 150 employees at a local high school (b) Lengths of 50 newly hatched pythons (c) Heights of 100 white pine trees in a forest (d) Amount of soda in 60 cups filled by an automated machine at a fast-food restaurant (e) Weights of 200 of the same candy bar in a shipment to a local supermarket 2. The proportion of observations from a standard Normal distribution that take values larger than 0.75 is about (a) 0.2266 (b) 0.7704 c) 0.7734 (d) 0.7764 (e) 0.8023 3. The density curve shown to the right takes the value 0.5 on the interval 0 x 2 and takes the value 0 everywhere else. What percent of the observations lie between 0.5 and 1.2? (a) 25% (b) 35% (c) 50% (d) 68% (e) 70% 0.5 0 1 2 4. The distribution of the heights of students in a large class is roughly Normal. Moreover, the average height is 68 inches, and approximately 95% of the heights are between 62 and 74 inches. Thus, the standard deviation of the height distribution is approximately equal to (a) 2 (b) 3 (c) 6 (d) 9 (e) 12 5. If a store runs out of advertised material during a sale, customers become upset, and the store loses not only the sale but also goodwill. From past experience, a music store finds that the mean number of CDs sold in a sale is 845, the standard deviation is 15, and a histogram of the demand is approximately Normal. The manager is willing to accept a 2.5% chance that a CD will be sold out. About how many CDs should the manager order for an upcoming sale? (a) 1295 (b) 1070 (c) 935 (d) 875 (e) 860 6. If your score on a test is at the 60th percentile, you know that your score lies (a) below the first quartile. (b) between the first quartile and the median. (c) between the median and the third quartile. (d) above the third quartile. (e) There is not enough information to say where it lies relative to the quartiles. 84 The Practice of Statistics, 4/e- Chapter 2 2011 BFW Publishers
7. In some courses (but certainly not in an intro stats course!), students are graded on a Normal curve. For example, students within ± 0.5 standard deviations of the mean receive a C; between 0.5 and 1.0 standard deviations above the mean receive a C+; between 1.0 and 1.5 standard deviations above the mean receive a B ; between 1.5 and 2.0 standard deviations above the mean receive a B, etc. The class average on an exam was 60 with a standard deviation of 10. The bounds for a B grade and the percent of students who will receive a B grade if the marks are actually Normally distributed are (a) (65, 75), 24.17% (b) (65, 75), 12.08% (c) (70, 75), 18.38% (d) (70, 75), 9.19% (e) (70, 75), 6.68% 8. The mean age (at inauguration) of all U.S. Presidents is approximately Normally distributed with a mean of 54.6. Barack Obama was 47 when he was inaugurated, which is the 11 th percentile of the distribution. Which of the following is closest to the standard deviation of presidents ages? (a) 9.20 (b) 6.18 (c) 6.18 (d) 7.60 (e) 9.20 9. Which of the following is not true about all Normal distributions? (a) The mean and median are equal (b) The points at which the curvature changes from up to down (the points of inflection) are one standard deviation away from the mean on either side. (c) About 2.5% of the values the variable takes on are more than two standard deviations above the mean. (d) About 68% of the values of the variable are more than one standard deviation away from the mean. (e) Z-scores of all the values of the variable have a mean of 0 and a standard deviation of 1. 10. The 16 th percentile of a Normally distributed variable has a value of 25 and the 97.5 th percentile has a value of 40. Which of the following is the best estimate of the mean and standard deviation of the variable? (a) Mean 32.5; Standard deviation 2.5 (b) Mean 32.5; Standard deviation 5 (c) Mean 32.5; Standard deviation 10 (d) Mean 30; Standard deviation 2.5 (e) Mean 30; Standard deviation 5 2011 BFW Publishers The Practice of Statistics, 4/e- Chapter 2 85
Part 2: Free Response Show all your work. Indicate clearly the methods you use, because you will be graded on the correctness of your methods as well as on the accuracy and completeness of your results and explanations. Raw scores on behavioral tests are often transformed for easier comparison. A test of reading ability has a mean of 75 and a standard deviation of 10 when given to third-graders. Sixthgraders have a mean score of 82 and a standard deviation of 11 on the same test. 11. David is a third-grade student who scores 78 on the test. Nancy is a sixth-grade student who scores 81. Calculate the z-score for each student. Who scored higher within his or her grade? 12. Suppose that the distribution of scores in each grade is Normal. Determine the percentiles for David and Nancy. Interpret your results in context. 13. A scientist is weighing each of 30 fish. She obtains a mean of 30 g and a standard deviation of 2 g. After completing the weighing, she finds that the scale was misaligned and always under reported every weight by 2 g that is, a fish that really weighed 26 g was reported to weigh 24 g. What is the mean and standard deviation after correcting for the error in the scale? 86 The Practice of Statistics, 4/e- Chapter 2 2011 BFW Publishers
14. On the density curve below, draw two vertical lines where you think the median and the mean of the distribution are. Label each line, and describe in words what feature of the curve you are using to locate each measure. Density X 15. Nitrates are organic compounds that are a substantial component of agricultural fertilizers. When those fertilizers run off into streams, the nitrates can have a toxic effect on animals that live in those streams. An ecologist studying nitrate pollution in two streams collects data on nitrate concentrations at 42 places on Stony Brook and 42 places on Mill Brook. The distribution for each stream is shown in the cumulative relative frequency graph below. Use this figure to compare the center and spread of nitrate concentrations in these two streams. Cumulative Relative Frequency 100 80 60 40 20 0 0 5 10 15 Nitrate concentration (mg/l) 20 Stony Mill 2011 BFW Publishers The Practice of Statistics, 4/e- Chapter 2 87
Test 2C Part 1 1. a Annual income in any company is likely to be skewed right, with upper level administrators on the right tail. The other four variables are more likely to be Normal. 2. c Percentile of z = 0.75 is.2266, so proportion above that value is 1.2266 =.7734. 3. b (1.2 0.5)(.5) =.35 or 35% 4. b By the 68-95-99.7 rule, 62 and 74 are both two standard deviations away from 68, so the standard deviation must be 3. 5. d By the 68-95-99.7 rule, about 2.5% of a Normal distribution is two standard deviations above the mean, and the standard deviation is 15. 2 x 15 = 30 points above the mean is 875. 6. c Median = 50 th percentile, Q 3 = 75 th percentile, so 60 th is between them. 7. d Scores that are 1 to 1.5 standard deviations above the mean earn a B, which means 10 to 15 points above 60, or 70 to 75. Area under the Standard Normal curve from z = 1 to z = 1.5 is.9332 -.8413 =.0919. 8. c 11 th 47 54.6 percentile corresponds to z = 1.23. 1.23, so s = 6.18. s 9. d About 68% of the values are less than one standard deviation away from the mean, not more. 10. e By the 68-95-99.7 rule, the 16 th percentile is about 1 standard deviation below the mean and the 97.5 th percentile is about 2 standard deviations above the mean. Hence the interval from 25 to 40 is three standard deviations, so the standard deviation is 5 and the mean is 25 + 5 = 30. Part 2 11. David s z-score = 78 75 0.3 ; Nancy s z-score = 81 82 0.09. David scored higher 10 11 within his grade. 12. z = 0.3 has a percentile of 0.6179, so David scored better than about 62 percent of other third-graders. z = 0.09 has a percentile of 0.4641, so Nancy scored better than about 46 percent of other sixth-graders. 13. Mean = 30 + 2 = 32 g.; the standard deviation is 2 g and is not affected by addition of a constant. 14. See below. The median (A) is located at a point such that there is an equal amount of area under the curve on each side. The mean (B) is located at the balance point of the curve. 15. Center of Mill Brook is higher: median for Mill Brook is about 8 mg/l and median for Stony Brook is about 5 mg/l. Mill Brook also has a greater spread than Stony Brook: IQR for Mill Brook is about 10 4 = 6 mg/l; IQR for Stony Brook is about 7.5 3 = 4.5 mg/l. Density A B X 2011 BFW Publishers The Practice of Statistics, 4/e- Chapter 2 93