2.5 Using the Mean and Standard Deviation to Describe Data

We’ve seen that if we are comparing the variability of two samples selected from a population, the sample with the larger standard deviation is the more variable of the two. Thus, we know how to interpret the standard deviation on a relative or comparative basis, but we haven’t explained how it provides a measure of variability for a single sample.

Teaching Tip

Use data collected in class to generate similar intervals and find the proportion of the class that falls into each interval.

To understand how the standard deviation provides a measure of variability of a data set, consider the following questions: How many measurements are within one standard deviation of the mean? How many measurements are within two standard deviations? For any particular data set, we can answer these questions by counting the number of measurement in each of the intervals. However, finding an answer that applies to any set of data, whether a population or a sample, is more problematic.

Rules 2.1 and 2.2 give two sets of answers to the questions of how many measurements fall within one, two, and three standard deviations of the mean. The first, which applies to any set of data, is derived from a theorem proved by the Russian mathematician P. L. Chebyshev (1821–1894). The second, which applies to mound-shaped, symmetric distributions of data (for which the mean, median, and mode are all about the same), is based upon empirical evidence that has accumulated over the years. However, the percentages given for the intervals in Rule 2.2 are not only more precise when the data are mound-shaped but provide remarkably good approximations even when the distribution of the data is slightly skewed or asymmetric. Note that the rules apply to either population or sample data sets.

Biography Pafnuty L. Chebyshev (1821–1894)

The Splendid Russian Mathematician

P. L. Chebyshev was educated in mathematical science at Moscow University, eventually earning his master’s degree. Following his graduation, Chebyshev joined St. Petersburg (Russia) University as a professor, becoming part of the well-known “Petersburg mathematical school.” It was here that he proved his famous theorem about the probability of a measurement being within k standard deviations of the mean (Rule 2.1). His fluency in French allowed him to gain international recognition in probability theory. In fact, Chebyshev once objected to being described as a “splendid Russian mathematician,” saying he surely was a “worldwide mathematician.” One student remembered Chebyshev as “a wonderful lecturer” who “was always prompt for class. As soon as the bell sounded, he immediately dropped the chalk and, limping, left the auditorium.”

Teaching Tip

Point out that Chebyshev’s rule gives the smallest percentages that are mathematically possible. In reality, the true percentages can be much higher than those stated.

Rule 2.1 Using the Mean and Standard Deviation to Describe Data: Chebyshev’s Rule

Chebyshev’s rule applies to any data set, regardless of the shape of the frequency distribution of the data.

  1. It is possible that very few of the measurements will fall within one standard deviation of the mean [i.e., within the interval (x¯−s, x¯+s) for samples and (μ−σ, μ+σ) for populations].

  2. At least 34 of the measurements will fall within two standard deviations of the mean [i.e., within the interval (x¯−2s, x¯+2s) for samples and (μ−2σ, μ+2σ) for populations].

  3. At least 89 of the measurements will fall within three standard deviations of the mean [i.e., within the interval (x¯−3s, x¯+3s) for samples and (μ−3σ, μ+3σ) for populations].

  4. Generally, for any number k greater than 1, at least (1−1/k2) of the measurements will fall within k standard deviations of the mean [i.e., within the interval (x¯−ks, x¯+ks) for samples and (μ−kσ, μ+kσ) for populations].

Rule 2.2 Using the Mean and Standard Deviation to Describe Data: The Empirical Rule

The empirical rule is a rule of thumb that applies to data sets with frequency distributions that are mound shaped and symmetric, as follows:

  1. Approximately 68% of the measurements will fall within one standard deviation of the mean [i.e., within the interval (x¯−s, x¯+s) for samples and (μ−σ, μ+σ) for populations].

  2. Approximately 95% of the measurements will fall within two standard deviations of the mean [i.e., within the interval (x¯−2s, x¯+2s) for samples and (μ−2σ, μ+2σ) for populations].

  3. Approximately 99.7% (essentially all) of the measurements will fall within three standard deviations of the mean [i.e., within the interval (x¯−3s, x¯+3s) for samples and (μ−3σ, μ+3σ) for populations].

Teaching Tip

Emphasize that the empirical rule applies only to mound-shaped, symmetric distributions.

Teaching Tip

Unlike the percentages associated with Chebyshev’s rule, the percentages presented in the empirical rule are only approximations. The real percentages could be higher or lower, depending on the data set analyzed.

Example 2.11 Interpreting the Standard Deviation—Rat-in-Maze Experiment

Problem

  1. Thirty students in an experimental psychology class use various techniques to train a rat to move through a maze. At the end of the course, each student’s rat is timed as it negotiates the maze. The results (in minutes) are listed in Table 2.7. Determine the fraction of the 30 measurements in the intervals x¯±s, x¯±2s, and x¯±3s, and compare the results with those predicted using Rules 2.1 and 2.2.

    Table 2.7 Times (in Minutes) of 30 Rats Running through a Maze

    Alternate View
    1.97  .60 4.02 3.20 1.15 6.06 4.44 2.02 3.37 3.65
    1.74 2.75 3.81 9.70 8.29 5.63 5.21 4.55 7.60 3.16
    3.77 5.36 1.06 1.71 2.47 4.25 1.93 5.15 2.06 1.65

    Data Set: RATMAZE

Solution

  1. First, we entered the data into the computer and used MINITAB to produce summary statistics. The mean and standard deviation of the sample data, highlighted on the printout shown in Figure 2.22, are

    [&*orule*{x}|=|3.74 ~rom~minutes|em||em|~normal~s|=|2.20 ~rom~minutes~normal~ &]

    x¯=3.74 minutess=2.20 minutes

    rounded to two decimal places.

    Figure 2.22

    MINITAB descriptive statistics for rat maze times

    Now we form the interval

    [&|pbo|*orule*{x}|-|s, *orule*{x}|+|s|pbc||=||pbo|3.74|-|2.20, 3.74|+|2.20|pbc||=||pbo|1.54, 5.94|pbc| &]

    (x¯−s, x¯+s)=(3.74−2.20, 3.74+2.20)=(1.54, 5.94)

    A check of the measurements shows that 23 of the times are within this one standard- deviation interval around the mean. This number represents 23/30, or ≈77%, of the sample measurements.

    The next interval of interest is

    [&|pbo|*orule*{x}|-|2s, *orule*{x}|+|2s|pbc||=||pbo|3.74|-|4.40, 3.74|+|4.40|pbc||=||pbo||minus|.66, 8.14|pbc| &]

    (x¯−2s, x¯+2s)=(3.74−4.40, 3.74+4.40)=(−.66, 8.14)

    All but two of the times are within this interval, so 28/30, or approximately 93%, are within two standard deviations of x¯.

    Finally, the three-standard-deviation interval around x¯ is

    [&|pbo|*orule*{x}|-|3s, *orule*{x}|+|3s|pbc||=||pbo|3.74|-|6.60, 3.74|+|6.60|pbc||=||pbo||minus|2.86, 10.34|pbc| &]

    (x¯−3s, x¯+3s)=(3.74−6.60, 3.74+6.60)=(−2.86, 10.34)

    All of the times fall within three standard deviations of the mean.

    These one-, two-, and three-standard-deviation percentages (77%, 93%, and 100%) agree fairly well with the approximations of 68%, 95%, and 100% given by the empirical rule (Rule 2.2).

Look Back

If you look at the MINITAB frequency histogram for this data set (Figure 2.23), you’ll note that the distribution is not really mound shaped, nor is it extremely skewed. Thus, we get reasonably good results from the mound-shaped approximations. Of course, we know from Chebyshev’s rule (Rule 2.1) that no matter what the shape of the distribution, we would expect at least 75% and at least 89% of the measurements to lie within two and three standard deviations of x¯, respectively.

Teaching Tip

It is helpful to students to use an example that demonstrates the differences in Chebyshev’s rule and the empirical rule. Emphasize the role that the symmetric distribution plays in determining the percentage of observations that fall into the tail of a distribution [e.g., above (x¯±2s)].

Figure 2.23

MINITAB histogram of rat maze times

Now Work Exercise 2.98

EPAGAS Example 2.12 Checking the Calculation of the Sample Standard Deviation

Problem

  1. Chebyshev’s rule and the empirical rule are useful as a check on the calculation of the standard deviation. For example, suppose we calculated the standard deviation for the gas mileage data (Table 2.2) to be 5.85. Are there any “clues” in the data that enable us to judge whether this number is reasonable?

Solution

  1. The range of the mileage data in Table 2.2 is 44.9−30.0=14.9. From Chebyshev’s rule and the empirical rule, we know that most of the measurements (approximately 95% if the distribution is mound shaped) will be within two standard deviations of the mean. And regardless of the shape of the distribution and the number of measurements, almost all of them will fall within three standard deviations of the mean. Consequently, we would expect the range of the measurements to be between 4 (i.e., ±2s) and 6 (i.e., ±3s) standard deviations in length. (See Figure 2.24.) For the car mileage data, this means that s should fall between

    [&*frac*{~rom~Range~normal~}{6}|=|~norm~*frac*{14.9}{6}|=|2.48 ~rom~*N*[1.5%0]and~normal~ *frac*{~rom~*N*[1.5%0]Range~normal~}{4}|=|*frac*{14.9}{4}|=|3.73~norm~ &]

    Range6=14.96=2.48 andRange4=14.94=3.73

    Hence, the standard deviation should not be much larger than 1/4 of the range, particularly for the data set with 100 measurements. Thus, we have reason to believe that the calculation of 5.85 is too large. A check of our work reveals that 5.85 is the variance s2, not the standard deviation s. (See Example 2.10.) We “forgot” to take the square root (a common error); the correct value is s=2.42. Note that this value is slightly smaller than the range divided by 6 (2.48). The larger the data set, the greater is the tendency for very large or very small measurements (extreme values) to appear, and when they do, the range may exceed six standard deviations.

    Figure 2.24

    The relation between the range and the standard deviation

Look Back

In examples and exercises, we’ll sometimes use s≈range/4 to obtain a crude, and usually conservatively large, approximation for s. However, we stress that this is no substitute for calculating the exact value of s when possible.

Now Work Exercise 2.99

In the next example, we use the concepts in Chebyshev’s rule and the empirical rule to build the foundation for making statistical inferences.

Example 2.13 Making a Statistical Inference—Car Battery Guarantee

Problem

A manufacturer of automobile batteries claims that the average length of life for its grade A battery is 60 months. However, the guarantee on this brand is for just 36 months. Suppose the standard deviation of the life length is known to be 10 months and the frequency distribution of the life-length data is known to be mound shaped.

  1. Approximately what percentage of the manufacturer’s grade A batteries will last more than 50 months, assuming that the manufacturer’s claim is true?

  2. Approximately what percentage of the manufacturer’s batteries will last less than 40 months, assuming that the manufacturer’s claim is true?

  3. Suppose your battery lasts 37 months. What could you infer about the manufacturer’s claim?

Solution

  1. If the distribution of life length is assumed to be mound shaped with a mean of 60 months and a standard deviation of 10 months, it would appear as shown in Figure 2.25. Note that we can take advantage of the fact that mound-shaped distributions are (approximately) symmetric about the mean, so that the percentages given by the empirical rule can be split equally between the halves of the distribution on each side of the mean.

    Figure 2.25

    Battery life-length distribution: manufacturer’s claim assumed true

    For example, since approximately 68% of the measurements will fall within one standard deviation of the mean, the distribution’s symmetry implies that approximately (1/2)(68%)=34% of the measurements will fall between the mean and one standard deviation on each side. This concept is illustrated in Figure 2.25. The figure also shows that 2.5% of the measurements lie beyond two standard deviations in each direction from the mean. This result follows from the fact that if approximately 95% of the measurements fall within two standard deviations of the mean, then about 5% fall outside two standard deviations; if the distribution is approximately symmetric, then about 2.5% of the measurements fall beyond two standard deviations on each side of the mean.

    1. It is easy to see in Figure 2.25 that the percentage of batteries lasting more than 50 months is approximately 34% (between 50 and 60 months) plus 50% (greater than 60 months). Thus, approximately 84% of the batteries should have a life exceeding 50 months.

    2. The percentage of batteries that last less than 40 months can also be easily determined from Figure 2.25: Approximately 2.5% of the batteries should fail prior to 40 months, assuming that the manufacturer’s claim is true.

    3. If you are so unfortunate that your grade A battery fails at 37 months, you can make one of two inferences: Either your battery was one of the approximately 2.5% that fail prior to 40 months, or something about the manufacturer’s claim is not true. Because the chances are so small that a battery fails before 40 months, you would have good reason to have serious doubts about the manufacturer’s claim. A mean smaller than 60 months or a standard deviation longer than 10 months would each increase the likelihood of failure prior to 40 months.*

Look Back

The approximations given in Figure 2.25 are more dependent on the assumption of a mound-shaped distribution than are the assumptions stated in the empirical rule (Rule 2.2) because the approximations in Figure 2.25 depend on the (approximate) symmetry of the mound-shaped distribution. We saw in Example 2.11 that the empirical rule can yield good approximations even for skewed distributions. This will not be true of the approximations in Figure 2.25; the distribution must be mound shaped and (approximately) symmetric.

Example 2.13 is our initial demonstration of the statistical inference-making process. At this point, you should realize that we’ll use sample information (in Example 2.13, your battery’s failure at 37 months) to make inferences about the population (in Example 2.13, the manufacturer’s claim about the length of life for the population of all batteries). We’ll build on this foundation as we proceed.

Statistics in Action Revisited

Interpreting Descriptive Statistics for the Body Image Data

We return to the analysis of the data from the Body Image: An International Journal of Research (Jan. 2010) study of 92 BDD patients. Recall that the quantitative variable of interest in the study is Total Appearance Evaluation score (ranging from 7 to 35 points). One of the questions of interest to the Brown University researchers was whether female BDD patients were less satisfied in their body image than normal females (i.e., females with no disorders), and whether male BDD patients were less satisfied in their body image than normal males. The analysis involved comparing the mean score for the BDD patients to the mean score for a normal group (called a “norm”). The appearance evaluation “norms” for females and males were 23.5 and 24.4, respectively.

MINITAB descriptive statistics for the data in the BDD file are displayed in Figure SIA2.5, with means and standard deviations highlighted. The sample mean score for females is 17.05 while the sample mean for males is 19.0. Note that both of these values fall well below the “norm” for each respective gender. Consequently, for this sample of BDD patients, both females and males have a lower average opinion about their body image than do normal females and males.

Figure SIA2.5

MINITAB descriptive statistics for appearance evaluation by gender

To interpret the standard deviations, we substitute into the formula, x¯±2s, to obtain the intervals:

Females: x¯±2s=17.05±2(4.76)=17.05±9.52=(7.53, 26.57)
Males: x¯±2s=19.00±2(5.42)=19.00±10.84=(8.16, 29.84)

From Chebyshev’s rule (Rule 2.1), we know that at least 75% of the females in the sample of BDD patients will have an Appearance Evaluation score anywhere between 7.53 and 26.57 points. Similarly, we know that at least 75% of the males in the sample of BDD patients will have an Appearance Evaluation score anywhere between 8.16 and 29.84 points. You can see that the interval for the females lies slightly below the interval for males. How does this information help in determining whether the female BDD patients and male BDD patients have mean scores below the “norm” for their gender? In Chapters 7 and 8 we will learn about an inferential method that uses both the sample mean and sample standard deviation to determine whether the population mean of the BDD patients is really smaller than the “norm” value.

Data Set: BDD

Exercises 2.93–2.113

Understanding the Principles

  1. 2.93 To what kind of data sets can Chebyshev’s rule be applied?

  2. 2.94 To what kinds of data sets can the empirical rule be applied?

  3. 2.95 The output from a statistical computer program indicates that the mean and standard deviation of a data set consisting of 200 measurements are $1,500 and $300, respectively.

    1. What are the units of measurement of the variable of interest? On the basis of the units, what type of data is this, quantitative or qualitative?

    2. What can be said about the number of measurements between $900 and $2,100? between $600 and $2,400? between $1,200 and $1,800? between $1,500 and $2,100?

  4. 2.96 For any set of data, what can be said about the percentage of the measurements contained in each of the following intervals?

    1. x¯−s to x¯+s

    2. x¯−2s to x¯+2s

    3. x¯−3s to x¯+3s

  5. 2.97 For a set of data with a mound-shaped relative frequency distribution, what can be said about the percentage of the measurements contained in each of the intervals specified in Exercise 2.96?

Learning the Mechanics

  1. L02098 2.98 The following is a sample of 25 measurements.

    Alternate View
    7  6 6 11 8 9 11  9 10  8 7 7 5
    9 10 7  7 7 7  9 12 10 10 8 6

    1. a. Compute x¯, s2, and s for this sample.

    2. b. Count the number of measurements in the intervals x¯±s, x¯±2s, and x¯±3s. Express each count as a percentage of the total number of measurements.

    3. c. Compare the percentages found in part b with the percentages given by the empirical rule and Chebyshev’s rule.

    4. d. Calculate the range and use it to obtain a rough approximation for s. Does the result compare favorably with the actual value for s found in part a?

  2. 2.99 Given a data set with a largest value of 760 and a smallest value of 135, what would you estimate the standard deviation to be? Explain the logic behind the procedure you used to estimate the standard deviation. Suppose the standard deviation is reported to be 25. Is this number reasonable? Explain. 

Applying the Concepts—Basic

  1. ROBOTS 2.100 Do social robots walk or roll? Refer to the International Conference on Social Robotics (Vol. 6414, 2010) study on the current trend in the design of social robots, Exercise 2.7 (p. 38). Recall that in a random sample of social robots obtained through a Web search, 28 were built with wheels. The numbers of wheels on each of the 28 robots are listed in the table on p. 77.

    1. Generate a histogram for the sample data set. Is the distribution of number of wheels mound shaped and symmetric?

    2. Find the mean and standard deviation for the sample data set.

    3. Form the interval x¯±2s.

    4. According to Chebychev’s rule, what proportion of sample observations will fall within the interval in part c?

    5. According to the empirical rule, what proportion of sample observations will fall within the interval in part c?

    6. Determine the actual proportion of sample observations that fall within the interval in part c. Even though the histogram in part a is not perfectly symmetric, does the empirical rule provide a good estimate of the proportion?

    Alternate View
    4 4 3 3 3 6 4 2 2 2 1 3 3 3
    3 4 4 3 2 8 2 2 3 4 3 3 4 2

    Source: Chew, S., et al. “Do social robots walk or roll?” International Conference on Social Robotics, Vol. 6414, 2010 (adapted from Figure 2).

  2. SANIT 2.101 Sanitation inspection of cruise ships. Refer to Exercise 2.41 (p. 51) and the Centers for Disease Control and Prevention listing of the sanitation scores for 186 cruise ships. The data are saved in the SANIT file.

    1. Find the mean and standard deviation of the sanitation scores.

    2. Calculate the intervals x¯±s, x¯±2s, and x¯±3s.

    3. Find the percentage of measurements in the data set that fall within each of the intervals in part b. Do these percentages agree with either Chebyshev’s rule or the empirical rule?

  3. 2.102 Motivation of drug dealers. Researchers at Georgia State University investigated the personality characteristics of drug dealers in order to shed light on their motivation for participating in the illegal drug market (Applied Psychology in Criminal Justice, Sept. 2009). The sample consisted of 100 convicted drug dealers who attended a court-mandated counseling program. Each dealer was scored on the Wanting Recognition (WR) Scale, which provides a quantitative measure of a person’s level of need for approval and sensitivity to social situations. (Higher scores indicate a greater need for approval.) The sample of drug dealers had a mean WR score of 39, with a standard deviation of 6. Assume the distribution of WR scores for drug dealers is mound shaped and symmetric.

    1. Give a range of WR scores that will contain about 95% of the scores in the drug dealer sample.

    2. What proportion of the drug dealers will have WR scores above 51?

    3. Give a range of WR sores that will contain nearly all the scores in the drug dealer sample.

  4. 2.103 Dentists’ use of anesthetics. A study published in Current Allergy & Clinical Immunology (Mar. 2004) investigated allergic reactions of dental patients to local anesthetics. Based on a survey of dental practitioners, the study reported that the mean number of units (ampoules) of local anesthetics used per week by dentists was 79, with a standard deviation of 23. Suppose we want to determine the percentage of dentists who use less than 102 units of local anesthetics per week.

    1. Assuming that nothing is known about the shape of the distribution for the data, what percentage of dentists use less than 102 units of local anesthetics per week?

    2. Assuming that the data have a mound-shaped distribution, what percentage of dentists use less than 102 units of local anesthetics per week?

  5. ISR 2.104 Irrelevant speech effects. Refer to the Acoustical Science & Technology (Vol. 35, 2014) study of irrelevant speech effects, Exercise 2.34 (p. 49). Recall that subjects performed a memorization task under two conditions: (1) with irrelevant background speech and (2) in silence. The difference in the error rates for the two conditions—called the relative difference in error rate (RDER)—was computed for each subject. Descriptive statistics for the RDER values are shown in the SAS printout.

    1. Based on the histogram shown in Exercise 2.34, which rule is more appropriate for describing the distribution of the RDER values? Why?

    2. Apply the rule you selected in part a to the data. Specifically, estimate the proportion of RDER values that fall in the interval x¯±2s.

  6. SUSTAIN 2.105 Corporate sustainability of CPA firms. Refer to the Business and Society (Mar. 2011) study on the sustainability behaviors of CPA corporations, Exercise 2.36 (pp. 50). Recall that the level of support for corporate sustainability (measured on a quantitative scale ranging from 0 to 160 points) was obtained for each 992 senior managers at CPA firms. Numerical measures of both central tendency and variation for level of support are shown in the MINITAB printout below. Give an estimate of the number of the 992 senior managers who have support levels between 41 and 95 points. (Use the histogram in Exercise 2.36 to help you decide which rule to apply.)

Applying the Concepts—Intermediate

  1. 2.106 Characteristics of antiwar demonstrators. The characteristics of antiwar demonstrators in the United States were examined in the American Journal of Sociology (Jan. 2014). Using data collected for over 5,000 antiwar demonstrators over a recent 3-year period, the researchers found that the mean number of protest organizations joined by the demonstrators was .90 with a standard deviation of 1.10.

    MINITAB Output for Exercise 2.105

    1. Give an interval that captures the number of protest organizations joined for at least 75% of the demonstrators.

    2. One of the antiwar demonstrators had joined 7 protest organizations. Is this typical of the sample of antiwar demonstrators? Explain.

  2. 2.107 Hand washing versus hand rubbing. As an alternative to hand washing, some hospitals allow health workers to rub their hands with an alcohol-based antiseptic. The British Medical Journal (Aug. 17, 2002) reported on a study to compare the effectiveness of washing the hands with soap and rubbing the hands with alcohol. One group of health care workers used hand rubbing, while a second group used hand washing to clean their hands. The bacterial count (number of colony-forming units) on the hand of each worker was recorded. The table gives descriptive statistics on bacteria counts for the two groups of health care workers.

    Mean Standard Deviation
    Hand rubbing 35  59
    Hand washing 69 106

    1. For hand rubbers, form an interval that contains at least 75% of the bacterial counts. (Note: The bacterial count cannot be less than 0.)

    2. Repeat part a for hand washers.

    3. On the basis of your results in parts a and b, make an inference about the effectiveness of the two hand-­cleaning methods.

  3. NZBIRDS 2.108 Extinct New Zealand birds. Refer to the Evolutionary Ecology Research (July 2003) study of the patterns of extinction in the New Zealand bird population, presented in Exercise 2.24(p. 42). Consider the data on the egg length (measured in millimeters) for the 132 bird species saved in the NZBIRDS file.

    1. Find the mean and standard deviation of the egg lengths.

    2. Form an interval that can be used to predict the egg length of a bird species found in New Zealand.

  4. 2.109 Velocity of Winchester bullets. The American Rifleman reported on the velocity of ammunition fired from the FEG P9R pistol, a 9-mm gun manufactured in Hungary. Field tests revealed that Winchester bullets fired from the pistol had a mean velocity (at 15 feet) of 936 feet per second and a standard deviation of 10 feet per second. Tests were also conducted with Uzi and Black Hills ammunition.

    1. Describe the velocity distribution of Winchester bullets fired from the FEG P9R pistol.

    2. A bullet whose brand is unknown is fired from the FEG P9R pistol. Suppose the velocity (at 15 feet) of the bullet is 1,000 feet per second. Is the bullet likely to be manufactured by Winchester? Explain.

  5. SAND 2.110 Permeability of sandstone during weathering. Refer to the Geographical Analysis (Vol. 42, 2010) study of the decay properties of sandstone when exposed to the weather, Exercises 2.69 and 2.90 (pp. 63 and 70). Recall that slices of sandstone blocks were measured for permeability under three conditions: no exposure to any type of weathering (A), repeatedly sprayed with a 10% salt solution (B), and soaked in a 10% salt solution and dried (C).

    1. Combine the mean (from Exercise 2.69) and standard deviation (from Exercise 2.90) to make a statement about where most of the permeability measurements for group A sandstone slices will fall. Which rule (and why) did you use to make this inference?

    2. Repeat part a for group B sandstone slices.

    3. Repeat part a for group C sandstone slices.

    4. Based on all your analyses, which type of weathering (type A, B, or C) appears to result in faster decay (i.e., higher permeability measurements)?

Applying the Concepts—Advanced

  1. 2.111 Shopping vehicle and judgment. While shopping at the grocery store, are you more likely to buy a vice product (e.g., a candy bar) when pushing a shopping cart or carrying a shopping basket? This was the question of interest in a study published in the Journal of Marketing Research (Dec. 2011). The researchers believe that when your arm is flexed (as when carrying a basket) you are more likely to choose a vice product than when your arm is extended (as when pushing a cart). To test this theory in a laboratory setting, the researchers recruited 22 consumers and had each push their hand against a table while they were asked a series of shopping questions. Half of the consumers were told to put their arm in a flex position (similar to a shopping basket), and the other half were told to put their arm in an extended position (similar to a shopping cart). Participants were offered several choices between a vice and a virtue (e.g., a movie ticket vs. a shopping coupon, pay later with a larger amount vs. pay now), and a choice score (on a scale of 0 to 100) was determined for each. (Higher scores indicate a greater preference for vice options.) The average choice score for consumers with a flexed arm was 59, while the average for consumers with an extended arm was 43.

    1. Suppose the standard deviations of the choice scores for the flexed arm and extended arm conditions are 4 and 2, respectively. Does this information support the researchers’ theory? Explain.

    2. Suppose the standard deviations of the choice scores for the flexed arm and extended arm conditions are 10 and 15, respectively. Does this information support the researchers’ theory? Explain.

  2. 2.112 Animal-assisted therapy for heart patients. A study was presented at the American Heart Association Conference (Nov. 2005) to gauge whether animal-assisted therapy can improve the physiological responses of heart failure patients. A team of nurses from the UCLA Medical Center randomly divided 76 heart patients into three groups. Each patient in group T was visited by a human volunteer accompanied by a trained dog, each patient in group V was visited by a volunteer only, and the patients in group C were not visited at all. The anxiety level of each patient was measured (in points) both before and after the visits. The table on p. 79 gives summary statistics for the drop in anxiety level for patients in the three groups. Suppose the anxiety level of a patient selected from the study had a drop of 22.5 points. From which group is the patient more likely to have come? Explain.

    Alternate View
    Sample Size Mean Drop Std. Dev.
    Group T: Volunteer +trained dog 26 10.5 7.6
    Group V: Volunteer only 25  3.9 7.5
    Group C: Control group (no visit) 25  1.4 7.5

    Based on Cole, K., et al. “Animal assisted therapy decreases hemodynamics, plasma epinephrine and state anxiety in hospitalized heart failure patients.” American Journal of Critical Care, 2007, 16: 575–585.

  3. 2.113 Land purchase decision. A buyer for a lumber company must decide whether to buy a piece of land containing 5,000 pine trees. If 1,000 of the trees are at least 40 feet tall, the buyer will purchase the land; otherwise, he won’t. The owner of the land reports that the height of the trees has a mean of 30 feet and a standard deviation of 3 feet. On the basis of this information, what is the buyer’s decision?

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