Joong Won Mo This set of WeBWorK problems is designed to orient you to the WeBWorK system and to help you learn how to communicate with the software. You will be learning about how to understand what you see on the screen and about how to enter your answers when you do the problems. You will practice entering numerical and functional expressions and look at ways to find and correct errors in your entries. WeBWorK assignment Homework2 is due on 06/26/2016 at 11:59pm PDT. 3. (1 pt) The summer monsoon brings 80% of India's rainfall and is essential for the country's agriculture. Records going back more than a century show that the amount of monsoon rainfall varies from year to year according to a distribution that is approximately Normal with mean 852 millimeters (mm) and standard deviation 82 mm. Use the 68-95-99.7 rule to answer the following questions. (a) Between what values do the monsoon rains fall in 95% of all years? (b) How small are the monsoon rains in the driest 2.5% of all years? mm to mm (a) from (b) less than mm 1. (1 pt) Examining the location of accidents on a level, 3mile bike path shows that they occur uniformly along the length of the path. This figure Answer(s) submitted: displays the density curve that describes the distribution of accidents. (a) The proportion of accidents that occur up to mile 1.6 of the path is the area under the density curve between 0 miles and 1.6 miles. What is this area? (b) Sue's property adjoins the bike path between the 1.6 mile mark and the 1.9 mile mark. What proportion of accidents happen in front of Sue's property? (a) (b) (incorrect) 4. (1 pt) In 2007, when she was a high school senior, Eleanor scored 650 on the mathematics part of the SAT. The distribution of SAT math scores in 2007 was Normal with mean 515 and standard deviation 114. Gerald took the ACT Assessment mathematics test and scored 27. ACT math scores for 2007 were Normally distributed with mean 21.0 and standard deviation 5.1. Find the standardized scores for both students. Eleanor's standardized score z = Gerald's standardized score z = Answer(s) submitted: (incorrect) Answer(s) submitted: 2. (1 pt) The times for the mile run of a large group of male college students are approximately Normal with mean 7.19 minutes and standard deviation 0.7 minutes. Use the 68-95-99.7 rule to answer the following questions. (Start by making a sketch of the density curve you can use to mark areas on.) (a) What range of times covers the middle 99.7% of this distribution? (b) What percent of these men run a mile in more than 6.49 minutes? minutes to minutes (a) from (b) % (incorrect) 5. (1 pt) The heights of women aged 20 to 29 are approximately Normal with mean 64 inches and standard deviation 2.7 inches. Men the same age have mean height 69.3 inches with standard deviation 2.8 inches. What are the z-scores for a woman 4'9\" tall and a man 6'4\" tall? (You may round your answers to two decimal places) z-scores for a woman 4'9\" tall: z-scores for a man 6'4\" tall: Answer(s) submitted: Answer(s) submitted: (incorrect) (incorrect) 1 9. (1 pt) Find the value z of a standard Normal variable that satisfies each of the following conditions. (a) The point z with 20% of the observations falling below it z= (b) The point z with 60% of the observations falling above it z= 6. (1 pt) Use Table A to find the proportion of the standard Normal distribution that satisfies each of the following statements. (a) z < 1.54 (b) z > 1.54 (c) z > 0.4 (d) 0.4 < z < 1.54 (a) (b) (c) (d) Answer(s) submitted: (incorrect) Answer(s) submitted: 10. (1 pt) The thorax lengths in a population of male fruit flies follow a Normal distribution with mean 0.785 millimeters (mm) and standard deviation 0.07 mm. What are the median and the first and third quartiles of thorax length? (a) Median: (b) The first quartile: (c) The third quartile: (incorrect) 7. (1 pt) The summer monsoon rains in India follow approximately a Normal distribution with mean 852 millimeters (mm) of rainfall and standard deviation 82 mm. Note: Use Table A to find the proportion or percentage below. (a) In the drought year 1987, 697 mm of rain fell. In what percent of all years will India have 697 mm or less of monsoon rain? (b) \"Normal rainfall\" means within 20% of the long-term average, or between 683 mm and 1022 mm. In what percent of all years is the rainfall normal? (a) % (b) % Answer(s) submitted: (incorrect) 11. (1 pt) Which of these variables is least likely to have a Normal distribution? A. Lengths of 50 newly hatched pythons B. Heights of 100 white pine trees in a forest C. Income per person for 150 different countries Answer(s) submitted: Answer(s) submitted: C (incorrect) (correct) 8. (1 pt) The common fruit fly Drosophila melanogaster is the most studied organism in genetic research because it is small, easy to grow, and reproduces rapidly. The length of the thorax (where the wings and legs attach) in a population of male fruit fies is approximately Normal with mean 0.800 millimeters (mm) and standard deviation 0.078 mm. Note: Use Table A to find the proportion below. (a) What proportion of flies have thorax length 0.765 mm or longer? (b) What proportion have thorax length between 0.765 mm and 0.98 mm? (a) (b) 12. (1 pt) It appears that people who are mildly obese are less active than leaner people. One study looked at the average number of minutes per day that people spending standing or walking. Among mildly obese people, minutes of activity varied according to the N(373, 67) distribution. Minutes of activity for lean people had the N(526, 107) distribution. Within what limits do the active minutes for 95% of the people in each group fall? Use the 68-95-99.7 rule. For mildly obese people: to mins to mins For lean people: Answer(s) submitted: Answer(s) submitted: (incorrect) (incorrect) 2 13. (1 pt) Scores on the Wechsler Adult Intelligence Scale (WAIS) are approximately Normal with mean 100 and standard deviation 15. People with WAIS score below 70 are considered mentally retarded when, for example, applying for Social Security disability benefits. According to the 68-95-99.7 rule, about what percent of adults are retarded by this criterion? % 17. (1 pt) Emissions of sulfur dioxide by industry set off chemical changes in the atmosphere that result in \"acid rain\". The acidity of liquids is measured by pH on a scale of 0 to 14. Distilled water has pH 7.0, and lower pH values indicate acidity. Normal rain is somewhat acidic, so acid rain is sometimes defined as rainfall with a pH below 5.0. The pH of rain at one location varies among rainy days according to a Normal distribution with mean 5.7 and standard deviation 0.58. What proportion of rainy days have rainfall with pH below 4.8? (Use Table A to find the proportion.) Answer(s) submitted: (incorrect) 14. (1 pt) Use Table A to find the proportion of observations from a standard Normal distribution that satisfies each of the following statements. (a) z -2.96 (b) z -2.96 (c) z 1.81 (d) -2.96 z 1.81 (a) (b) (c) (d) Answer(s) submitted: (incorrect) 18. (1 pt) Automated manufacturing operations are quite precise but still vary, often with distribution that are close to Normal. The width in inches of slots cut by a milling machine follows approximately the N(0.92, 0.0008) distribution. The specifications allow slot widths between 0.91975 and 0.92025. What proportion of slots meet these specifications? Answer as a percent: %. Answer(s) submitted: Answer(s) submitted: (incorrect) (incorrect) 19. (1 pt) A pharmaceutical manufacturer forms tablets by compressing a granular material that contains the active ingredient and various fillers. The force in kilograms (kg) applied to the tablets varies a bit, with the N (11.7, 0.2) distribution. The process specifications call for applying a force between 11.4 and 12.4 kg. (a) What percent of tablets are subject to a force that meets the specifications? % (b) The manufacturer adjusts the process so that the mean force is at the center of the specifications, = 11.9 kg. The standard deviation remains 0.2 kg. What percent now meet the specifications? % 15. (1 pt) Use the value of from Table A that comes closest to satisfying the condition. (a) Find the number z such that the proportion of observations that are less than z in a standard Normal distribution is 0.282. (b) Find the number z such that 89.05% of all observations from a standard Normal distribution are greater than z. (a) (b) Answer(s) submitted: (incorrect) Answer(s) submitted: 16. (1 pt) After the physical training required during World War II, the distribution of mile run times for male students at the University of Illinois was approximately Normal with mean 7.11 minutes and standard deviation 0.74 minutes. What proportion of these students could run a mile in 6 minutes or less? (incorrect) 20. (1 pt) In its Fuel Economy Guide for 2008 model vehicles, the Environmental Protection Agency gives data on 1152 vehicles. There are a number of outliers, mainly vehicles with very poor gas mileage. If we ignore the outliers, however, the combined city and highway gas mileage of the other 1120 or so vehicles is approximately Normal with mean 18.7 miles per gallon (mpg) and standard deviation 4.3 mpg. Answer(s) submitted: (incorrect) 3 The 2008 Chevrolet Malibu with a four-cylinder engine has combined gas mileage 25 mpg. What percent of all vehicles have worse gas mileage than the Malibu? % How high must a 2008 vehicle's gas mileage be in order to fall in the top 15% of all vehicles? (The distribution omits a few high outliers, mainly hybrid gas-electric vehicles.) 23. (1 pt) The heights of women aged 20 to 29 follow approximately the N(64, 2.65) distribution. Men the same age have heights distributed as N(69.3, 2.75). What percent of young women are shorter than the mean height of young men? % Answer(s) submitted: The quartiles of any distribution are the values with cumulative proportions 0.25 and 0.75. They span the middle half of the distribution. What are the quartiles of the distribution of gas mileage? Gas mileage of the first quartile: Gas mileage of the third quartile: The quintiles of any distribution are the values with cumulative proportions 0.20, 0.40, 0.60, and 0.80. What are the quintiles of the distribution of gas mileage? Gas mileage of 0.20 of the distribution: Gas mileage of 0.40 of the distribution: Gas mileage of 0.60 of the distribution: Gas mileage of 0.80 of the distribution: (incorrect) 24. (1 pt) Changing the mean and standard deviation of a Normal distribution by a moderate amount can greatly change the percent of observation in the tails. Suppose that a college is looking for applicants with SAT math scores 730 and above. (a) In 2007, the scores of men on the math SAT followed the N(533, 116) distribution. What percent of men scored 730 or better? (b) Women's SAT math scores that year had the N(499, 110) distribution. What percent of women scored 730 or better? % (a) (b) % Answer(s) submitted: Answer(s) submitted: (incorrect) 25. (1 pt) Osteoporosis is a condition in which the bones become brittle due to loss of minerals. Bone mineral density (BMD) is measured to diagnose osteoporosis. BMD is often reported in standard Normal form. The standardization is based on a population of healthy young adults. The World Health Organization (WHO) criterion for osteoporosis is a BMD 2.5 standard deviations below the mean for young adults. BMD measurements in a population of people similar in age and sex roughly follow a Normal distribution. (a) What percent of healthy young adults have osteoporosis by the WHO criterion? (b) Mean BMD for women ages 70-79 is about 2 standard deviations below the mean for young adults. Suppose that the standard deviation for these older women is the same as for young adults. What percent of this older population have osteoporosis (that is, BMD at least 2.5 standard deviations below that of young adults)? (a) % (b) % (incorrect) 21. (1 pt) Reports on a student's ACT or SAT usually give the percentile as well as the actual score. The percentile is just the cumulative proportion stated as a percent: the percent of all scores that were lower than this one. In 2007, composite ACT scores were close to Normal with mean 21.2 and standard deviation 5.0. Jacob scored 24.5. What was his percentile? (Round to the nearest whole percent.) Jacob's percentile: % Answer(s) submitted: (incorrect) 22. (1 pt) It is possible to score higher than 1600 on the SAT, but scores 1600 and above are reported as 1600. In 2007, the distribution of SAT scores (combining mathematics and reading) was close to Normal with mean 1021 and standard deviation 211. What proportion of 2007 SAT scores were reported as 1600? Proportion: Answer(s) submitted: Answer(s) submitted: (incorrect) (incorrect) 4 (c) the percent of observations from the N(21.2, 5.0) distribution that are greater than 27. (The percent greater than or equal to 27 is the same, because the area exactly over 27 is 0.) (a) % (b) % (c) % 26. (1 pt) Scores on the ACT tests for the 2007 high school graduating class had mean 21.2 and standard deviation 5.0. In all, 1,300,599 students in this class took the test. Of these, 149,164 had scores higher than 27 and another 50,310 had scores exactly 27. The genuine Normal N(21.2,5.0) distribution includes all values, but ACT scores are always whole numbers. Because of this, the distribution of ACT scores is only approximately normal. To illustrate this fact, find (a) the percent of 2007 ACT scores greater than 27. (b) the percent of 2007 ACT scores greater than or equal to 27. Answer(s) submitted: (incorrect) c Generated by WeBWorK, http://webwork.maa.org, Mathematical Association of America 5