The quartiles of any distribution are the values with cumulative proportions 0.25and

Sketch density curves. Sketch density curves that describe distributions with the following shapes: (a) Symmetric, but with two peaks (that is, two strong clusters of observations). (b) Single peak and skewed to the left.

A uniform distribution. Figure 3.6 displays the density curve of a uniform distribution. The curve takes the constant value 1 over the interval from 0 to 1 and is zero outside that range of values. This means that data described by this distribution take values that are uniformly spread between 0 and 1. Use areas under this density curve to answer the following questions. (a) Why is the total area under this curve equal to 1? (b) What percent of the observations lie above 0.8? (c) What percent of the observations lie below 0.6? (d) What percent of the observations lie between 0.25 and 0.75?

Mean and median. What is the mean of the density curve pictured in Figure 3.6? What is the median?

Mean and median. Figure 3.7 displays three density curves, each with three points marked on them. At which of these points on each curve do the mean and the median fall?

Heights of young women. The distribution of heights of women aged 20 to 29 is approximately Normal with mean 64 inches and standard deviation 2.7 inches.2 Draw a Normal curve on which this mean and standard deviation are correctly located. (Hint: Draw the curve first, locate the points where the curvature changes, then mark the horizontal axis.)

Heights of young women. The distribution of heights of women aged 20 to 29 is approximately Normal with mean 64 inches and standard deviation 2.7 inches. Use the 689599.7 rule to answer the following questions. (Start by making a sketch like Figure 3.10.) (a) Between what heights do the middle 95% of young women fall? (b) What percent of young women are taller than 61.3 inches?

Length of pregnancies. The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. Use the 689599.7 rule to answer the following questions. (a) Between what values do the lengths of almost all (99.7%) pregnancies fall? (b) How short are the shortest 2.5% of all pregnancies?

SAT versus ACT. Eleanor scores 680 on the mathematics part of the SAT. The distribution of SAT math scores in recent years has been Normal with mean 518 and standard deviation 114. Gerald takes the ACT Assessment mathematics test and scores 27. ACT math scores are Normally distributed with mean 20.7 and standard deviation 5.0. Find the standardized scores for both students. Assuming that both tests measure the same kind of ability, who has the higher score?

Mens and womens heights. 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 6 feet tall and a man 6 feet tall? Say in simple language what information the z-scores give that the actual heights do not.

How hard do locomotives pull? An important measure of the performance of a locomotive is its adhesion, which is the locomotives pulling force as a multiple of its weight. The adhesion of one 4400-horsepower diesel locomotive model varies in actual use according to a Normal distribution with mean = 0.37 and standard deviation = 0.04. (a) What proportion of adhesions measured in use are higher than 0.40? (b) What proportion of adhesions are between 0.40 and 0.50?

A better locomotive. Improvements in the locomotives computer controls change the distribution of adhesion to a Normal distribution with mean = 0.41 and standard deviation = 0.02. Find the proportions in (a) and (b) of the previous exercise after this improvement.

Table A. Use Table A to find the value z of a standard Normal variable that satisfies each of the following conditions. (Use the value of z from Table A that comes closest to satisfying the condition.) In each case, sketch a standard Normal curve with your value of z marked on the axis. (a) The point z with 25% of the observations falling below it. (b) The point z with 40% of the observations falling above it.

IQ test scores. Scores on the Wechsler Adult Intelligence Scale are approximately Normally distributed with = 100 and = 15. (a) What scores fall in the lowest 25% of the distribution? (b) How high a score is needed to be in the highest 5%?

Which of these variables is least likely to have a Normal distribution? (a) Income per person for 150 different countries (b) Lengths of 50 newly hatched pythons (c) Heights of 100 white pine trees in a forest

To completely specify the shape of a Normal distribution, you must give (a) the mean and the standard deviation. (b) the five-number summary. (c) the mean and the median.

Figure 3.14 shows a Normal curve. The mean of this distribution is (a) 0. (b) 2. (c) 3.

The standard deviation of the Normal distribution in Figure 3.14 is (a) 2. (b) 3. (c) 5.

The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. 95% of all pregnancies last between (a) 250 and 282 days. (b) 234 and 298 days. (c) 218 and 314 days.

The scores of adults on an IQ test are approximately Normal with mean 100 and standard deviation 15. Corinne scores 118 on such a test. Her z-score is about (a) 1.2. (b) 7.87. (c) 18.

The proportion of observations from a standard Normal distribution that take values less than 1.15 is about (a) 0.1251. (b) 0.8531. (c) 0.8749.

The proportion of observations from a standard Normal distribution that take values larger than 0.75 is about (a) 0.2266. (b) 0.7734. (c) 0.8023.

The scores of adults on an IQ test are approximately Normal with mean 100 and standard deviation 15. Corinne scores 118 on such a test. She scores higher than what percent of all adults? (a) About 12% (b) About 88% (b) About 98%

Understanding density curves. Remember that it is areas under a density curve, not the height of the curve, that give proportions in a distribution. To illustrate this, sketch a density curve that has its peak at 0 on the horizontal axis but has greater area within 0.25 on either side of 1 than within 0.25 on either side of 0.

Are the data Normal? Soil penetrability. Table 2.3 (page 61) gives data on the penetrability of soil at each of three levels of compression. We might expect the penetrability of specimens of the same soil at the same level of compression to follow a Normal distribution. Make stemplots of the data for loose and for intermediate compression. Does either sample seem roughly Normal? Does either appear distinctly non-Normal? If so, what kind of departure from Normality does your stemplot show?

IQ test scores. The Wechsler Adult Intelligence Scale (WAIS) is the most common IQ test. The scale of scores is set separately for each age group and is approximately Normal with mean 100 and standard deviation 15. According to the 689599.7 rule, about what percent of people have WAIS scores (a) above 100? (b) above 145? (c) below 85?

Low IQ test scores. Scores on the Wechsler Adult Intelligence Scale (WAIS) are approximately Normal with mean 100 and standard deviation 15. People with WAIS scores below 70 are considered mentally retarded when, for example, applying for Social Security disability benefits. According to the 689599.7 rule, about what percent of adults are retarded by this criterion?

Actual IQ test scores. Here are the IQ test scores of 31 seventh-grade girls in a Midwest school district:4 114 100 104 89 102 91 114 114 103 105 108 130 120 132 111 128 118 119 86 72 111 103 74 112 107 103 98 96 112 112 93 (a) We expect IQ scores to be approximately Normal. Make a stemplot to check that there are no major departures from Normality. (b) Nonetheless, proportions calculated from a Normal distribution are not always very accurate for small numbers of observations. Find the mean x and standard deviation s for these IQ scores. What proportions of the scores are within one standard deviation and within two standard deviations of the mean? What would these proportions be in an exactly Normal distribution?

Standard Normal drill. (a) Find the number z such that the proportion of observations that are less than z in a standard Normal distribution is 0.8. (b) Find the number z such that 35% of all observations from a standard Normal distribution are greater than z. ACT versus SAT. There are two major tests of readiness for college: the ACT and the SAT. ACT scores are reported on a scale from 1 to 36. The distribution of ACT scores in recent years has been roughly Normal with mean = 20.9 and standard deviation = 4.8. SAT scores are reported on a scale from 400 to 1600. SAT scores have been roughly Normal with mean = 1026 and standard deviation = 209. Exercises 3.32 to 3.43 are based on this information.

Tonya scores 1318 on the SAT. Jermaine scores 27 on the ACT. Assuming that both tests measure the same thing, who has the higher score?

Jacob scores 16 on the ACT. Emily scores 670 on the SAT. Assuming that both tests measure the same thing, who has the higher score?

Jos e scores 1287 on the SAT. Assuming that both tests measure the same thing, what score on the ACT is equivalent to Jos es SAT score?

Maria scores 28 on the ACT. Assuming that both tests measure the same thing, what score on the SAT is equivalent to Marias ACT score?

Reports on a students 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. Tonya scores 1318 on the SAT. What is her percentile?

Reports on a students 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. Jacob scores 16 on the ACT. What is his percentile?

It is possible to score higher than 1600 on the SAT, but scores 1600 and above are reported as 1600. What proportion of SAT scores are reported as 1600?

It is possible to score higher than 36 on the ACT, but scores 36 and above are reported as 36. What proportion of ACT scores are reported as 36?

How well must Abigail do on the ACT in order to place in the top 20% of all students?

The quartiles of any distribution are the values with cumulative proportions 0.25 and 0.75. What are the quartiles of the distribution of ACT scores?

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 SAT scores?

Heights of men and women. The heights of women aged 20 to 29 follow approximately the N(64, 2.7) distribution. Men the same age have heights distributed as N(69.3, 2.8). What percent of young women are taller than the mean height of young men?

Heights of men and women. The heights of women aged 20 to 29 follow approximately the N(64, 2.7) distribution. Men the same age have heights distributed as N(69.3, 2.8). What percent of young men are shorter than the mean height of young women?

A surprising calculation. Changing the mean of a Normal distribution by a moderate amount can greatly change the percent of observations in the tails. Suppose that a college is looking for applicants with SAT math scores 750 and above. (a) In 2004, the scores of men on the math SAT followed the N(537, 116) distribution. What percent of men scored 750 or better? (b) Womens SAT math scores that year had the N(501, 110) distribution. What percent of women scored 750 or better? You see that the percent of men above 750 is almost three times the percent of women with such high scores. Why this is true is controversial.

Grading managers. Many companies grade on a bell curve to compare the performance of their managers and professional workers. This forces the use of some low performance ratings so that not all workers are listed as above average. Ford Motor Companys performance management process for a time assigned 10% A grades, 80% B grades, and 10% C grades to the companys 18,000 managers. Suppose that Fords performance scores really are Normally distributed. This year, managers with scores less than 25 received Cs and those with scores above 475 received As. What are the mean and standard deviation of the scores?

Osteoporosis. Osteoporosis is a condition in which the bones become brittle due to loss of minerals. To diagnose osteoporosis, an elaborate apparatus measures bone mineral density (BMD). BMD is usually reported in standardized 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) Women aged 70 to 79 are of course not young adults. The mean BMD in this age is about 2 on the standard scale for young adults. Suppose that the standard deviation is the same as for young adults. What percent of this older population has osteoporosis?

Are the data Normal? ACT scores. Scores on the ACT test for the 2004 high school graduating class had mean 20.9 and standard deviation 4.8. In all, 1,171,460 students in this class took the test, and 1,052,490 of them had scores of 27 or lower.5 If the distribution of scores were Normal, what percent of scores would be 27 or lower? What percent of the actual scores were 27 or lower? Does the Normal distribution describe the actual data well?

How accurate is 689599.7? The 689599.7 rule for Normal distributions is APPLET a useful approximation. To see how accurate the rule is, drag one flag across the other so that the applet shows the area under the curve between the two flags. (a) Place the flags one standard deviation on either side of the mean. What is the area between these two values? What does the 689599.7 rule say this area is? (b) Repeat for locations two and three standard deviations on either side of the mean. Again compare the 689599.7 rule with the area given by the applet.

Where are the quartiles? How many standard deviations above and below the APPLET mean do the quartiles of any Normal distribution lie? (Use the standard Normal distribution to answer this question.)

Grading managers. In Exercise 3.47, we saw that Ford Motor Company grades APPLET its managers in such a way that the top 10% receive an A grade, the bottom 10% a C, and the middle 80% a B. Lets suppose that performance scores follow a Normal distribution. How many standard deviations above and below the mean do the A/B and B/C cutoffs lie? (Use the standard Normal distribution to answer this question.)