#61-71, 85-89

Section 2.2 Exercises 143 Normal states? The Normal probability plot displays data on the areas (in thousands of square miles) of each of the 50 states. Use the graph to determine if the distribution of land area is approximately Normal. Expected z-score (a) The median is at the yellow line and the mean is at the red line. 200 400 600 Area (thousands of miz) (b) The median is at the red line and the mean is at the yellow line. 2. Density of the earth In 1798, the English scientist Henry Cavendish measured the density of the earth several (c) The mode is at the red line and the median is at the times by careful work with a torsion balance. The variable yellow line. recorded was the density of the earth as a multiple of the (d) The mode is at the yellow line and the median is at the density of water. A Normal probability plot of the data is red line. shown. 16 Use the graph to determine if this distribution of density measurement is approximately Normal. (e) The mode is at the red line and the mean is at the yellow line. 2 - Exercises 86-88 refer to the following setting. The weights of laboratory cockroaches can be modeled with a Normal 1 - . ... . . distribution having mean 80 grams and standard deviation 2 grams. The following figure is the Normal curve for this Expected z-score 0 - .. . .. . . distribution of weights. -1 - -2 - 5.0 5.2 5.4 5.6 5 .8 Density of earth $3. Refrigerators Refer to Exercise 75. a) Use your calculator to make a Normal probability plot of the data. Sketch this graph on your paper. b) What does the graph in part (a) imply about whether B E F G the distribution of refrigerator capacity is approximately Normal? Explain. 86. Point C on this Normal curve corresponds to 84. Big sharks Refer to Exercise 76. (a) 84 grams. (b) 82 grams. (c) 78 grams. a) Use your calculator to make a Normal probability plot (d) 76 grams. (e) 74 grams. of the data. Sketch this graph on your paper. 87. About what percent of the cockroaches have weights (6) What does the graph in part (a) imply about whether between 76 and 84 grams? the distribution of shark length is approximately (a) 99.7% (b) 95%% (c) 68% Normal? Explain. (d) 47.5% (e) 34% Multiple Choice: Select the best answer for Exercises 85-90. 88. About what proportion of the cockroaches will have weights greater than 83 grams? 5. Two measures of center are marked on the density curve shown. Which of the following is (a) 0.0228 (b) 0.0668 (c) 0.1587 correct? (d) 0.9332 (e) 0.0772MODELING DISTRIBUTIONS OF QUANTITATIVE DATA 142 CHAPTER 2 Based on the relationship between the mean, standard 75. Refrigerators Consumer Reports magazine deviation, minimum, and maximum, is it reasonable g $32 collected data on the usable capacity (in cubic to believe that the distribution of Michigan tuitions is feet) of a sample of 36 side-by-side refrigerators. approximately Normal? Explain your answer. Here are the data: 78. Are body weights Normal? The heights of people 12.9 13.7 14.1 14.2 14.5 14.5 14.6 14.7 15.1 15.2 15.3 15.3 of the same gender and similar ages follow Normal 15.3 15.3 15.5 15.6 15.6 15.8 16.0 16.0 16.2 16.2 16.3 16.4 distributions reasonably closely. How about body 16.5 16.6 16.6 16.6 16.8 17.0 17.0 17.2 17.4 17.4 17.9 18.4 weights? The weights of women aged 20 to 29 have mean 141.7 pounds and median 133.2 pounds. A histogram of the data and summary statistics The first and third quartiles are 1 18.3 pounds and are shown here. Is this distribution of refrigerator 157.3 pounds. Is it reasonable to believe that the capacities approximately Normal? Justify your distribution of body weights for women aged 20 to 29 ; answer based on the graph and the empirical rule. approximately Normal? Explain your answer. 79. Runners' heart rates The following figure is a Normal 12 - g 135 probability plot of the heart rates of 200 male runners 10 - after 6 minutes of exercise on a treadmill. '+ Use the graph to determine if this distribution of heart rates is Frequency approximately Normal. 2 - w 12 13 14 15 16 17 18 19 2 - Usable capacity (ft3) n Mean SD Min Q Med Q3 Max Expected z-score 0 - 36 15.825 1.217 12.9 15.15 15.9 16.6 18.4 -1 - 76. Big sharks Here are the lengths (in feet) of 44 great white sharks: 13 - 2 - 3 18.7 12.3 18.6 16.4 15.7 18.3 14.6 15.8 14.9 17.6 12.1 16.4 16.7 17.8 16.2 12.6 17.8 13.8 12.2 15.2 14.7 12.4 70 80 90 100 110 120 130 140 150 13.2 15.8 14.3 16.6 9.4 18.2 13.2 13.6 15.3 16.1 13.5 Heart rate (beats per minute) 19.1 16.2 22.8 16.8 13.6 13.2 15.7 19.7 18.7 13.2 16.8 80. Carbon dioxide emissions The following figure is a A dotplot of the data and summary statistics are Normal probability plot of the emissions of carbon shown below. Is this distribution of shark length dioxide (CO2) per person in 48 countries. 15 Use the approximately Normal? Justify your answer based graph to determine if this distribution of CO2 emissions on the graph and the empirical rule. is approximately Normal. 10 12 3 - 14 16 18 20 22 Length (ft) 2 - n Mean SD Min Q Med Q3 Max 1 - 14 15.586 2.55 9.4 13.55 15.75 17.2 22.8 Expected z-score 0 _ 77. Is Michigan Normal? We collected data on the tuition -1 - charged by colleges and universities in Michigan. Here - 2 are some numerical summaries for the data: Mean SD Min - 3 Max 10,614 8049 1873 30,823 0 2 4 6 8 10 12 14 16 18 20 CO2 emissions (metric tons per person)Section 2.2 Exercises b) MENSA is an elite organization that admits as members people who score in the top 2% on IQ tests. (c) Which of the two options in parts (a) and (b) do you What score on the Wechsler Adult Intelligence Scale think is preferable? Justify your answer. (Be sure to would an individual aged 20 to 34 have to earn to consider the effect of these changes on the percent of qualify for MENSA membership? lids that are too large to fit.) 65. Post office A local post office weighs outgoing 71. Flight times An airline flies the same route at mail and finds that the weights of first-class letters the same time each day. The flight time varies are approximately Normally distributed with a according to a Normal distribution with unknown mean of 0.69 ounce and a standard deviation of mean and standard deviation. On 15% of days, the 0.16 ounce . flight takes more than an hour. On 3% of days, the a) Estimate the 60th percentile of first-class letter flight lasts 75 minutes or more. Use this information to determine the mean and standard deviation of the weights . flight time distribution. (b ) First-class letters weighing more than 1 ounce require extra postage. What proportion of first-class letters at 72. Brush your teeth The amount of time Ricardo spends brushing his teeth follows a Normal this post office require extra postage? distribution with unknown mean and standard Exercises 69 and 70 refer to the following setting. At some deviation. Ricardo spends less than 1 minute brushing fast-food restaurants, customers who want a lid for their his teeth about 40% of the time. He spends more than drinks get them from a large stack near the straws, napkins, 2 minutes brushing his teeth 2% of the time. Use this information to determine the mean and standard and condiments. The lids are made with a le with a small amount of deviation of this distribution. flexibility so they can be stretched across the mouth of the cup and then snugly secured. When lids are too small or too 73. Normal highway driving? The dotplot shows the large, customers can get very frustrated, especially if they end EPA highway gas mileage estimates in miles per gallon (mpg) for a random sample of 21 model up spilling their drinks. At one particular restaurant, large year 2018 midsize cars. " Explain why this drink cups require lids with a "diameter" of between 3.95 distribution of highway gas mileage is not and 4.05 inches. The restaurant's lid supplier claims that the approximately Normal. diameter of its large lids follows a Normal distribution with mean 3.98 inches and standard deviation 0.02 inch. Assume that the supplier's claim is true. 69. Put a lid on it! 25 30 (a) What percent of large lids are too small to fit? Highway gas mileage (mpg) (b) What percent of large lids are too big to fit? (c) Compare your answers to parts (a) and (b). Does it 74. Normal to be foreign born? The histogram displays make sense for the lid manufacturer to try to make the percent of foreign-born residents in each of the one of these values larger than the other? Why or 50 states. " Explain why this distribution of the why not? percent of foreign-born residents in the states is not 70. Put a lid on it! The supplier is considering two approximately Normal. changes to reduce to 1% the percentage of its large- cup lids that are too small. One strategy is to adjust 20 the mean diameter of its lids. Another option is to alter the production process, thereby decreasing the standard deviation of the lid diameters. umber of states a If the standard deviation remains at o = 0.02 inch, 10 at what value should the supplier set the mean diameter of its large-cup lids so that only 1% are too small to fit? If the mean diameter stays at ( = 3.98 inches, what value of the standard deviation will result in only 1% of 10 15 30 lids that are too small to fit? Percent foreign-born residentsMODELING DISTRIBUTIONS OF QUANT 144 CHAPTER 2 Recycle and Review 89. A different species of cockroach has weights that are 91. Making money (2.1 approximately Normally distributed with a mean of 50 total family income grams. After measuring the weights of many of these from Indiana (38 inc cockroaches, a lab assistant reports that 14% of the (44 individuals). Me cockroaches weigh more than 55 grams. Based on this given below the dot report, what is the approximate standard deviation of weight for this species of cockroach? (a) 4.6 (b) 5.0 (c) 6.2 Indiana ... . (d) 14.0 (e) Cannot determine without more ... .. ..... information. ::::. New Jersey 90. The following Normal probability plot shows the 0 25,000 50,0 distribution of points scored for the 551 players in a Tot single NBA season. New Expected z-score WNHOTNW Consider individuals in e of $95,000. Which indivi to others in his or her stat support your answer. 500 1000 1500 2000 Points 92. More money (1.3) R (a) How do the ranges o If the distribution of points was displayed in a Justify your answer. histogram, what would be the best description of the histogram's shape? (b) Explain why the stan (a) Approximately Normal incomes in the New than for the Indiana (b) Symmetric but not approximately Normal (c) Skewed left (d) Skewed right (e) Cannot be determinedMODELING DISTRIBUTIONS OF QUANTITATIVE DATA 140 CHAPTER 2 60. Standard Normal areas Find the proportion of observations in a standard Normal distribution that 52. Batter up! Refer to Exercise 48. Use the empirical rule satisfies each of the following statements. to answer the following questions. About what percent of Major League Baseball players (a) 2 -1.66 Intelligence Scale (an IQ test) for the 20- to 34-year-old age group are approximately Normally distributed with (b) - 1.66