The following data give the lengths of time to failure for

Problem 1E For each of the following situations, identify the population of interest, the inferential objective, and how you might go about collecting a sample. a A university researcher wants to estimate the proportion of U.S. citizens from “Generation X” who are interested in starting their own businesses. b For more than a century, normal body temperature for humans has been accepted to be 98.6° Fahrenheit. Is it really? Researchers want to estimate the average temperature of healthy adults in the United States. c A city engineer wants to estimate the average weekly water consumption for single-family dwelling units in the city. d The National Highway Safety Council wants to estimate the proportion of automobile tires with unsafe tread among all tires manufactured by a specific company during the current production year. e Apolitical scientist wants to determine whether a majority of adult residents of a state favor a unicameral legislature. f A medical scientist wants to estimate the average length of time until the recurrence of a certain disease. g An electrical engineer wants to determine whether the average length of life of transistors of a certain type is greater than 500 hours.

In Exercise 1.19, the mean and standard deviation of the amount of chloroform present in water sources were given to be 34 and 53, respectively. You argued that the amounts of chloroform could therefore not be normally distributed. Use Tchebysheff’s theorem (Exercise 1.32) to describe the distribution of chloroform amounts in water sources.

Are some cities more windy than others? Does Chicago deserve to be nicknamed “The Windy City”? Given below are the average wind speeds (in miles per hour) for 45 selected U.S. cities: 8.9 12.4 8.6 11.3 9.2 8.8 35.1 6.2 7.0 7.1 11.8 10.7 7.6 9.1 9.2 8.2 9.0 8.7 9.1 10.9 10.3 9.6 7.8 11.5 9.3 7.9 8.8 8.8 12.7 8.4 7.8 5.7 10.5 10.5 9.6 8.9 10.2 10.3 7.7 10.6 8.3 8.8 9.5 8.8 9.4 a Construct a relative frequency histogram for these data. (Choose the class boundaries without including the value 35.1 in the range of values.) b The value 35.1 was recorded at Mt. Washington, New Hampshire. Does the geography of that city explain the magnitude of its average wind speed? c The average wind speed for Chicago is 10.3 miles per hour. What percentage of the cities have average wind speeds in excess of Chicago’s? d Do you think that Chicago is unusually windy?

Of great importance to residents of central Florida is the amount of radioactive material present in the soil of reclaimed phosphate mining areas. Measurements of the amount of \({ }^{238} \mathrm{U}\) in 25 soil samples were as follows (measurements in picocuries per gram): .74 6.47 1.90 2.69 .75 .32 9.99 1.77 2.41 1.96 1.66 .70 2.42 .54 3.36 3.59 .37 1.09 8.32 4.06 4.55 .76 2.03 5.70 12.48 Construct a relative frequency histogram for these data. Equation Transcription: Text Transcription: ^238U

Given here is the relative frequency histogram associated with grade point averages (GPAs) of a sample of 30 students: a Which of the GPA categories identified on the horizontal axis are associated with the largest proportion of students? b What proportion of students had GPAs in each of the categories that you identified? c What proportion of the students had GPAs less than 2.65?

The top 40 stocks on the over-the-counter (OTC) market, ranked by percentage of outstanding shares traded on one day last year are as follows: 11.88 6.27 5.49 4.81 4.40 3.78 3.44 3.11 2.88 2.68 7.99 6.07 5.26 4.79 4.05 3.69 3.36 3.03 2.74 2.63 7.15 5.98 5.07 4.55 3.94 3.62 3.26 2.99 2.74 2.62 7.13 5.91 4.94 4.43 3.93 3.48 3.20 2.89 2.69 2.61 a Construct a relative frequency histogram to describe these data. b What proportion of these top 40 stocks traded more than 4% of the outstanding shares? c If one of the stocks is selected at random from the 40 for which the preceding data were taken, what is the probability that it will have traded fewer than 5% of its outstanding shares?

The relative frequency histogram given next was constructed from data obtained from a random sample of 25 families. Each was asked the number of quarts of milk that had been purchased the previous week. a Use this relative frequency histogram to determine the number of quarts of milk purchased by the largest proportion of the 25 families. The category associated with the largest relative frequency is called the modal category. b What proportion of the 25 families purchased more than 2 quarts of milk? c What proportion purchased more than 0 but fewer than 5 quarts?

The self-reported heights of 105 students in a biostatistics class were used to construct the histogram given below. a Describe the shape of the histogram. b Does this histogram have an unusual feature? c Can you think of an explanation for the two peaks in the histogram? Is there some consid- eration other than height that results in the two separate peaks? What is it?

An article in Archaeometry presented an analysis of 26 samples of Romano–British pottery, found at four different kiln sites in the United Kingdom. The percentage of aluminum oxide in each of the 26 samples is given below: a Construct a relative frequency histogram to describe the aluminum oxide content of all 26 pottery samples. b What unusual feature do you see in this histogram? Looking at the data, can you think of an explanation for this unusual feature?

Problem 9E Resting breathing rates for college-age students are approximately normally distributed with mean 12 and standard deviation 2.3 breaths per minute. What fraction of all college-age students have breathing rates in the following intervals? a 9.7 to 14.3 breaths per minute b 7.4 to 16.6 breaths per minute c 9.7 to 16.6 breaths per minute d Less than 5.1 or more than 18.9 breaths per minute

The following results on summations will help us in calculating the sample variance s2. For any constant c, a \(\sum_{i=1}^{n} c=n c\). b \(\sum_{i=1}^{n} c y_{i}=c \sum_{i=1}^{n} y_{i}\). c \(\sum_{i=1}^{n}\left(x_{i}+y_{i}\right)=\sum_{i=1}^{n} x_{i}+\sum_{i=1}^{n} y_{i}\). Use (a), (b), and (c) to show that \(s^{2}=\frac{1}{n-1} \sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2}=\frac{1}{n-1}\left[\sum_{i=1}^{n} y_{i}^{2}-\frac{1}{n}\left(\sum_{i=1}^{n} y_{i}\right)^{2}\right]\). Equation Transcription: Text Transcription: s^2 sum i=1nc=nc sum i=1ncy_i=c sumi=1ny_i sum i=1n(x_i+y_i)=sum i=1nx_i+sum i=1ny_i s^2=1 over n-1 sum i=1n(yi-y bar)^2=1 over n-1[sum i=1ny_i^2-1 over n (sum i=1ny_i)^2]

Use the result of Exercise 1.11 to calculate s for the \(n=6\) sample measurements 1, 4, 2, 1, 3, and 3. Equation Transcription: Text Transcription: n=6

Problem 10E It has been projected that the average and standard deviation of the amount of time spent online using the Internet are, respectively, 14 and 17 hours per person per year (many do not use the Internet at all!). a What value is exactly 1 standard deviation below the mean? b If the amount of time spent online using the Internet is approximately normally distributed, what proportion of the users spend an amount of time online that is less than the value you found in part (a)? c Is the amount of time spent online using the Internet approximately normally distributed? Why?

Refer to Exercise a Calculate \(\bar{y}\) and for the data given. b Calculate the interval \(\bar{y} \pm k s\) for \(k=1\), 2 and 3. Count the number of measurements that fall within each interval and compare this result with the number that you would expect according to the empirical rule. Equation Transcription: Text Transcription: y bar y bar+/-ks k=1

Refer to Exercise 1.3 and repeat parts (a) and (b) of Exercise 1.13.

The range of a set of measurements is the difference between the largest and the smallest values. The empirical rule suggests that the standard deviation of a set of measurements may be roughly approximated by one-fourth of the range (that is, range/4). Calculate this approximation to s for the data sets in Exercises 1.2, 1.3, and 1.4. Compare the result in each case to the actual, calculated value of s.

Refer to Exercise 1.4 and repeat parts (a) and (b) of Exercise 1.13.

Problem 19E According to the Environmental Protection Agency, chloroform, which in its gaseous form is suspected to be a cancer-causing agent, is present in small quantities in all the country’s 240,000 public water sources. If the mean and standard deviation of the amounts of chloroform present in water sources are 34 and 53 micrograms per liter (?g/L), respectively, explain why chloroform amounts do not have a normal distribution.

Problem 20E Weekly maintenance costs for a factory, recorded over a long period of time and adjusted for inflation, tend to have an approximately normal distribution with an average of $420 and a standard deviation of $30. If $450 is budgeted for next week, what is an approximate probability that this budgeted figure will be exceeded?

Problem 21E The manufacturer of a new food additive for beef cattle claims that 80% of the animals fed a diet including this additive should have monthly weight gains in excess of 20 pounds. A large sample of measurements on weight gains for cattle fed this diet exhibits an approximately normal distribution with mean 22 pounds and standard deviation 2 pounds. Do you think the sample information contradicts the manufacturer’s claim? (Calculate the probability of a weight gain exceeding 20 pounds.)

Problem 23SE The mean duration of television commercials is 75 seconds with standard deviation 20 seconds. Assume that the durations are approximately normally distributed to answer the following. a What percentage of commercials last longer than 95 seconds? b What percentage of the commercials last between 35 and 115 seconds? c Would you expect commercial to last longer than 2 minutes? Why or why not?

In Exercise , there is one extremely large value . Eliminate this value and calculate \(\bar{y}\) and for the remaining 39 observations. Also, calculate the intervals \(\bar{y} \pm k s\) for \(k=1\), 2, and 3; count the number of measurements in each; then compare these results with those predicted by the empirical rule. Compare the answers here to those found in Exercise Note the effect of a single large observation on \(\bar{y}\) and . Equation Transcription: Text Transcription: y bar y bar +/- ks k=1 y bar

Aqua running has been suggested as a method of cardiovascular conditioning for injured athletes and others who desire a low-impact aerobics program. In a study to investigate the relationship between exercise cadence and heart rate,1 the heart rates of 20 healthy volunteers were measured at a cadence of 48 cycles per minute (a cycle consisted of two steps). The data are as follows: 87 109 79 80 96 95 90 92 96 98 101 91 78 112 94 98 94 107 81 96 a Use the range of the measurements to obtain an estimate of the standard deviation. b Construct a frequency histogram for the data. Use the histogram to obtain a visual approximation to \(\bar{y}\) and . c Calculate \(\bar{y}\) and . Compare these results with the calculation checks provided by parts (a) and (b). d Construct the intervals \(\bar{y} \pm k s\), \(k=1\), 2, and 3, and count the number of measurements falling in each interval. Compare the fractions falling in the intervals with the fractions that you would expect according to the empirical rule. Equation Transcription: Text Transcription: ^1 y bar y bar y bar +/- ks k=1

Prove that the sum of the deviations of a set of measurements about their mean is equal to zero; that is, \(\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)=0\). Equation Transcription: Text Transcription: sum i=1n(y_i-y bar)=0

Problem 28SE The discharge of suspended solids from a phosphate mine is normally distributed with mean daily discharge 27 milligrams per liter (mg/L) and standard deviation 14 mg/L. In what proportion of the days will the daily discharge be less than 13 mg/L?

The College Board’s verbal and mathematics Scholastic Aptitude Tests are scored on a scale of 200 to 800. It seems reasonable to assume that the distribution of test scores are approximately normally distributed for both tests. Use the result from Exercise 1.17 to approximate the standard deviation for scores on the verbal test.

The following data give the lengths of time to failure for \(n=88\) radio transmitter-receivers: 16 224 16 80 96 536 400 80 392 576 128 56 656 224 40 32 358 384 256 246 328 464 448 716 304 16 72 8 80 72 56 608 108 194 136 224 80 16 424 264 156 216 168 184 552 72 184 240 438 120 308 32 272 152 328 480 60 208 340 104 72 168 40 152 360 232 40 112 112 288 168 352 56 72 64 40 184 264 96 224 168 168 114 280 152 208 160 176 a Use the range to approximate for the \(n=88\) lengths of time to failure. b Construct a frequency histogram for the data. [Notice the tendency of the distribution to tail outward (skew) to the right.] c Use a calculator (or computer) to calculate \(\bar{y}) and . (Hand calculation is much too tedious for this exercise.) d Calculate the intervals \(\bar{y} \pm k s\), \(k=1\), 2 and 3, and count the number of measurements falling in each interval. Compare your results with the empirical rule results. Note that the empirical rule provides a rather good description of these data, even though the distribution is highly skewed. Equation Transcription: Text Transcription: n=88 n=88 y bar y bar+/-ks k=1

Problem 29SE A machine produces bearings with mean diameter 3.00 inches and standard deviation 0.01 inch. Bearings with diameters in excess of 3.02 inches or less than 2.98 inches will fail to meet quality specifications. a Approximately what fraction of this machine’s production will fail to meet specifications? b What assumptions did you make concerning the distribution of bearing diameters in order to answer this question?

A set of 340 examination scores exhibiting a bell-shaped relative frequency distribution has a mean of \(\bar{y}=72\) and a standard deviation of \(s=8\). Approximately how many of the scores would you expect to fall in the interval from 64 to 80 ? The interval from 56 to 88 ? Equation Transcription: Text Transcription: y bar=72 s=8

Problem 30SE Compared to their stay-at-home peers, women employed outside the home have higher levels of high-density lipoproteins (HDL), the “good” cholesterol associated with lower risk for heart attacks. A study of cholesterol levels in 2000 women, aged 25–64, living in Augsburg,Germany, was conducted by Ursula Haertel, Ulrigh Keil, and colleagues2 at the GSF-Medis Institut in Munich. Of these 2000women, the 48%whoworked outside the home had HDL levels that were between 2.5 and 3.6 milligrams per deciliter (mg/dL) higher than the HDL levels of their stayat- home counterparts. Suppose that the difference in HDL levels is normally distributed, with mean 0 (indicating no difference between the two groups of women) and standard deviation 1.2 mg/dL. If you were to select an employed woman and a stay-at-home counterpart at random, what is the probability that the difference in their HDL levels would be between 1.2 and 2.4?

Compare the ratio of the range to s for the three sample sizes (\(n=6\), 20, and 88) for Exercises 1.12, 1.24, and 1.25. Note that the ratio tends to increase as the amount of data increases. The greater the amount of data, the greater will be their tendency to contain a few extreme values that will inflate the range and have relatively little effect on s. We ignored this phenomenon and suggested that you use 4 as the ratio for finding a guessed value ofs in checking calculations. Equation Transcription: Text Transcription: n=6

Problem 31SE Over the past year, a fertilizer production process has shown an average daily yield of 60 tons with a variance in daily yields of 100. If the yield should fall to less than 40 tons tomorrow, should this result cause you to suspect an abnormality in the process? (Calculate the probability of obtaining less than 40 tons.) What assumptions did you make concerning the distribution of yields?

Let \(k \geq 1\) . Show that, for any set of \(n\) measurements, the fraction included in the interval \(y^{-}-k s\) to \(y^{-}+k s\) is at least \(\left(1-1 / k^{2}\right)\). [Hint:\(s^{2}=\frac{1}{n-1}\left[\sum_{i=1}^{n}\left(y_{i}-y^{-}\right)^{2}\right]\) In this expression, replace all deviations for which \(\left|y_{i}-y^{-}\right| \geq k s\) with \(ks\). Simplify.] This result is known as \(T\) chebysheff's theorem. Equation Transcription: Text Transcription: k geq 1 n y^- -ks y^- +ks (1-1/k^2) s^2=1/n-1[sum_i=1^n(y_i-y^-)^2 |y_i-y^-|geq ks ks T

A personnel manager for a certain industry has records of the number of employees absent per day. The average number absent is 5.5, and the standard deviation is 2.5. Because there are many days with zero, one, or two absent and only a few with more than ten absent, the frequency distribution is highly skewed. The manager wants to publish an interval in which at least 75% of these values lie. Use the result in Exercise 1.32 to find such an interval.

A pharmaceutical company wants to know whether an experimental drug has an effect on systolic blood pressure. Fifteen randomly selected subjects were given the drug and, after sufficient time for the drug to have an impact, their systolic blood pressures were recorded. The data appear below: 172 140 123 130 115 148 108 129 137 161 123 152 133 128 142 a Approximate the value of s using the range approximation. b Calculate the values of \(\bar{y}\) and s for the 15 blood pressure readings. c Use Tchebysheff’s theorem (Exercise 1.32) to find values a and b such that at least 75% of the blood pressure measurements lie between a and b. d Did Tchebysheff’s theorem work? That is, use the data to find the actual percent of blood pressure readings that are between the values a and b you found in part (c). Is this actual percentage greater than 75%? Equation Transcription: Text Transcription: y bar

Problem 34SE For the data discussed in Exercise 1.33, give an upper bound to the fraction of days when there are more than 13 absentees. Reference A personnel manager for a certain industry has records of the number of employees absent per day. The average number absent is 5.5, and the standard deviation is 2.5. Because there are many days with zero, one, or two absent and only a few with more than ten absent, the frequency distribution is highly skewed. The manager wants to publish an interval in which at least 75% of these values lie. Use the result in Exercise 1.32 to find such an interval.

Problem 37SE Studies indicate that drinking water supplied by some old lead-lined city piping systems may contain harmful levels of lead. Based on data presented by Karalekas and colleagues,4 it appears that the distribution of lead content readings for individual water specimens has mean .033 mg/L and standard deviation .10 mg/L. Explain why it is obvious that the lead content readings are not normally distributed.

A random sample of 100 foxes was examined by a team of veterinarians to determine the prevalence of a specific parasite. Counting the number of parasites of this specific type, the veterinarians found that 69 foxes had no parasites of the type of interest, 17 had one parasite of the 3. Exercises preceded by an asterisk are optional. type under study, and so on. A summary of their results is given in the following table: Number of Parasites 0 1 2 3 4 5 6 7 8 Number of Foxes 69 17 6 3 1 2 1 0 1 a Construct the relative frequency histogram for the number of parasites per fox. b Calculate \(\bar{y}\) and s for the data given. c What fraction of the parasite counts falls within 2 standard deviations of the mean? Within 3 standard deviations? Do your results agree with Tchebysheff’s theorem (Exercise 1.32) and/or the empirical rule? Equation Transcription: Text Transcription: y bar