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descriptive statistics
Seeing Through Statistics 3rd Edition David D Busch, Jessica M Utts - Solutions
9. Explain how two variables can have a perfect curved relationship and yet have zero correlation. Draw a picture of a set of data meeting those criteria.
8. Give an example of a pair of variables that are likely to have a positive correlation and a pair of variables that are likely to have a negative correlation.
*7. Which implies a stronger linear relationship, a correlation of .4 or a correlation of .6? Explain.
6. The relationship between height and weight is a well-established and obvious fact. Suppose you were to sample heights and weights for a small number of your friends, and you failed to find a statistically significant relationship between the two variables. Would you conclude that the
5. Suppose a weak relationship exists between two variables in a population.Which would be more likely to result in a statistically significant relationship between the two variables: a sample of size 100 or a sample of size 10,000?Explain.
4. Are each of the following pairs of variables likely to have a positive correlation or a negative correlation?a. Daily temperatures at noon in New York City and in Boston measured for a year.b. Weights of automobiles and their gas mileage in average miles per gallon.c. Hours of television watched
*3. A pint of water weighs 1.04 pounds, so 1 pound of water is 0.96 pint. Suppose a merchant sells water in containers weighing 0.5 pound, but customers can fill them to their liking. It is easier to weigh the filled container than to measure the volume of water the customer is purchasing. Define x
2. In Figure 10.2, we observed that the correlation between husbands’ and wives’heights, measured in millimeters, was .36. Can you determine what the correlation would be if the heights were converted to inches (and not rounded off)?Explain.
*1. Suppose 100 different researchers each did a study to see if there was a relationship between coffee consumption and height. Suppose there really is no such relationship in the population. Would you expect any of the researchers to find a statistically significant relationship? If so,
3. Find some data that represent change over time for a topic of interest to you.Present a line graph of the data in the best possible format. Explain what you have done to make sure the picture is as useful as possible.
2. Collect two measurement variables on each of at least 10 individuals. Represent them in a statistical picture. Describe the picture in terms of possible outliers, variability, and relationship between the two variables.
1. Collect some categorical data on a topic of interest to you and represent it in a statistical picture. Explain what you have done to make sure the picture is as useful as possible.
17. Refer to Figure 2 on page 691 of Original Source 11 on the CD, “Driving impairment due to sleepiness is exacerbated by low alcohol intake.”a. What type of picture is in the figure?b. Write a few sentences explaining what you learn from the picture about subjective sleepiness ratings under
*16. Refer to Figure 1 on page 691 of Original Source 11 on the CD, “Driving impairment due to sleepiness is exacerbated by low alcohol intake.”*a. What type of picture is in the figure?b. Write a few sentences explaining what you learn from the picture about lane drifting episodes under the
15. Refer to Additional News Story 19 on the CD, a press release from Cornell University entitled “Puppy love’s dark side: First study of love-sick teens reveals higher risk of depression, alcohol use and delinquency.” The article includes a graph labeled “Adjusted change in depression
14. Table 9.5 provides the total number of men and women who were employed in 1971, 1981, 1991, and 2001 in the United States.a. Create a bar graph for the data.b. Compare the bar graph to the one in Figure 9.2, which presents the percent of men and women who were employed. Discuss what can be
13. According to an article in The Seattle Times (Meckler, 2003), living organ donors are most often related to the organ recipient. Table 9.4 gives the percentages of each type of relationship for all 6613 cases where an organ was transplanted from a living donor in 2002 in the United States.
*12. Find a graph that does not start at zero. Redraw the picture to start at zero. Discuss the pros and cons of the two versions.
11. Find an example of a statistical picture in a newspaper or magazine that has at least one of the problems listed in Section 9.4, “Difficulties and Disasters in Plots, Graphs, and Pictures.” Explain the problem. If you think anything should have been done differently, explain what and why.
10. According to the American Medical Association Family Medical Guide (1982, p. 422), the distribution of blood types in the United States in the late 1970s was as shown in Table 9.3.a. Draw a pie chart illustrating the blood-type distribution for white Americans, ignoring the RH factor.b. Draw a
9. Find an example of a statistical picture in a newspaper or magazine or on the Internet.Answer the 10 questions in Section 9.5 for the picture. In the process of answering the questions, explain what (if any) features you think should have been added or changed to make it a good picture. Include
*8. Table 9.2 indicates the population (in millions) and the number of violent crimes(in millions) in the United States from 1982 to 1991, as reported in the World Almanac and Book of Facts (1993, p. 948).a. Draw two line graphs representing the trend in violent crime over time. Draw the first
7. Figure 10.4 in Chapter 10 displays the success rate for professional golfers when putting at various distances. Discuss the figure in the context of the material in this chapter. Are there ways in which the picture could be improved?
6. In its February 24–26, 1995 edition (p. 7), USA Weekend gave statistics on the changing status of which parent children live with. As noted in the article, the numbers don’t total 100% because they are drawn from two sources: the U.S.Census Bureau and America’s Children: Resources from
*5. Figure 9.10, which displays rising postal rates, is an example of a graph with misleading units because the prices are not adjusted for inflation. The graph actually has another problem as well. Use the checklist in Section 9.5 to determine the problem; then redraw the graph correctly (but
4. An article in Science (23 January 1998, 279, p. 487) reported on a “telephone survey of 2600 parents, students, teachers, employers, and college professors”in which people were asked the question, “Does a high school diploma mean that a student has at least learned the basics?” Results
3. One method used to compare authors or to determine authorship on unsigned writing is to look at the frequency with which words of different lengths appear in a piece of text. For this exercise, you are going to compare your own writing with that of the author of this book.a. Using the first full
2. Suppose a real estate company in your area sold 100 houses last month, whereas their two major competitors sold 50 houses and 25 houses, respectively. The top company wants to display its better record with a pictogram using a simple twodimensional picture of a house. Draw two pictograms
*1. Give the name of a type of statistical picture that could be used for each of the following kinds of data:*a. One categorical variable*b. One measurement variable*c. Two categorical variables*d. Two measurement variables
25. Math SAT scores for students admitted to a university are bell-shaped with a mean of 520 and a standard deviation of 60.a. Draw a picture of these SAT scores, indicating the cutoff points for the middle 68%, 95%, and 99.7% of the scores.b. A student had a math SAT score of 490. Find the
*24. Over many years, rainfall totals for Sacramento, CA in January ranged from a low of about 0.05 inch to a high of about 19.5 inches. The median was about 3.1 inches. Based on this information, explain how you can tell that the distribution of rainfall values in Sacramento in January cannot be
23. Recall that GRE scores are approximately bell-shaped with a mean of 497 and standard deviation of 115. The minimum and maximum possible scores on the GRE exam are 200 and 800, respectively.a. What is the range for GRE scores?b. Refer to the result about the relationship between the range and
22. Suppose that you were told that scores on an exam in a large class you are taking ranged from 50 to 100 and that they were approximately bell-shaped.a. Estimate the mean for the exam scores.b. Refer to the result about the relationship between the range and standard deviation in the previous
21. Remember from Chapter 7 that the range for a data set is found as the difference between the maximum and minimum values. Explain why it makes sense that for a bell-shaped data set of a few hundred values, the range should be about 4 to 6 standard deviations.
*20. According to Chance magazine ([1993], 6, no. 3, p. 5), the mean healthy adult temperature is around 98.2° Fahrenheit, not the previously assumed value of 98.6°. Suppose the standard deviation is 0.6 degree and the population of healthy temperatures is bell-shaped.*a. What proportion of the
19. Assuming heights for each sex are bell-shaped, with means of 70 inches for men and 65 inches for women, and with standard deviations of 3 inches for men and 2.5 inches for women, what proportion of your sex is shorter than you are? (Be sure to mention your sex and height in your answer!)
*18. Suppose you record how long it takes you to get to work or school over many months and discover that the times are approximately bell-shaped with a mean of 15 minutes and a standard deviation of 2 minutes. How much time should you allow to get there to make sure you are on time 90% of the time?
17. Suppose a candidate for public office is favored by only 48% of the voters. If a sample survey randomly selects 2500 voters, the percentage in the sample who favor the candidate can be thought of as a measurement from a normal curve with a mean of 48% and a standard deviation of 1%. Based on
16. For every 100 births in the United States, the number of boys follows, approximately, a normal curve with a mean of 51 boys and standard deviation of 5 boys.If the next 100 births in your local hospital resulted in 36 boys (and thus 64 girls), would that be unusual? Explain.
*15. Recall that for Stanford-Binet IQ scores the mean is 100 and the standard deviation is 16.*a. Use the Empirical Rule to specify the ranges into which 68%, 95%, and 99.7% of Stanford-Binet IQ scores fall.b. Draw a picture similar to Figure 8.5 for Stanford-Binet scores, illustrating the ranges
14. A graduate school program in English will admit only students with GRE verbal ability scores in the top 30%. What is the lowest GRE score it will accept?(Recall the mean is 497 and the standard deviation is 115.)
13. Give an example of a population of measurements that you do not think has a normal curve, and draw its frequency curve.
*12. Recall from Chapter 7 that the interquartile range covers the middle 50% of the data. For a bell-shaped population:*a. The interquartile range covers what range of standardized scores? In other words, what are the standardized scores for the lower and upper quartiles?(Hint: Draw a standard
11. Use Table 8.1 to verify that the Empirical Rule is true. You may need to round off the values slightly.
*10. Every time you have your cholesterol measured, the measurement may be slightly different due to random fluctuations and measurement error. Suppose that for you the population of possible cholesterol measurements if you are healthy has a mean of 190 and a standard deviation of 10. Further,
9. Mensa is an organization that allows people to join only if their IQs are in the top 2% of the population.a. What is the lowest Stanford-Binet IQ you could have and still be eligible to join Mensa? (Remember that the mean is 100 and the standard deviation is 16.)b. Mensa also allows members to
*8. Find the percentile for the observed value in the following situations:*a. GRE score of 450 (mean 497, s.d. 115).b. Stanford-Binet IQ score of 92 (mean 100, s.d. 16).c. Woman’s height of 68 inches (mean 65 inches, s.d. 2.5 inches).
7. Draw a picture of a bell-shaped curve with a mean value of 100 and a standard deviation of 10. Mark the mean and the intervals derived from the Empirical Rule in the appropriate places on the horizontal axis. You do not have to mark the vertical axis. Use Figure 8.5 as a guide.
6. The 84th percentile for the Stanford-Binet IQ test is 116. (Recall that the mean is 100 and the standard deviation is 16.)a. Verify that this is true by computing the standardized score and using Table 8.1.b. Draw pictures of the original and standardized scores to illustrate this situation,
5. Using Table 8.1, a computer, or a calculator, determine the percentage of the population falling between the two standard scores given:a. 1.00 and 1.00b. 1.28 and 1.75c. 0.0 and 1.00
*4. Using Table 8.1, a computer, or a calculator, determine the standard score that has the following percentage of the population above it:*a. 2%b. 50%*c. 75%d. 10%
3. Using Table 8.1, a computer, or a calculator, determine the standard score that has the following percentage of the population below it:a. 25%b. 75%c. 45%d. 98%
*2. Using Table 8.1, a computer, or a calculator, determine the percentage of the population falling above each of the following standard scores:*a. 1.28*b. 0.25*c. 2.33
1. Using Table 8.1, a computer, or a calculator, determine the percentage of the population falling below each of the following standard scores:a. 1.00b. 1.96c. 0.84
2. Measure your pulse rate 25 times over the next few days, but don’t take more than one measurement in any 10-minute period. Record any unusual events related to the measurements, such as if one was taken during exercise or one was taken immediately upon awakening. Create a stemplot and a
1. Find a set of data that has meaning for you. Some potential sources are the Internet, the sports pages, and the classified ads. Using the methods given in this chapter, summarize and display the data in whatever ways are most useful. Give a written description of interesting features of the data.
*35. Refer to Original Source 20 on the CD, “Organophosphorus pesticide exposure of urban and suburban preschool children with organic and conventional diets.”In Table 4 on page 381, information is presented for estimated dose levels of various pesticides for children who eat organic versus
34. Refer to Original Source 5 on the CD, “Distractions in everyday driving.” Notice that on page 86 of the report, when the responses are summarized for the quantitative data in Question 8, only the mean is provided. But for Questions 7 and 9 the mean and median are provided. Why do you think
33. According to the National Weather Service, there is about a 10% chance that total annual rainfall for Sacramento, CA will be less than 11.1 inches and a 20%chance that it will be less than 13.5 inches. At the upper end, there is about a 10% chance that it will exceed 29.8 inches and a 20%
32. The Winters [CA] Express on October 30, 2003, reported that the seasonal rainfall(since July 1) for the year was 0.36 inches, and that the “Normal to October 28 rainfall is 1.14 inches. Does this mean that the area received abnormally low rainfall in the period from July 1 to October 28,
31. Refer to the data in Exercise 29. Using the definition of outliers on page 135, identify which value(s) are outliers in each of the two sets of values (Males and Females).
*30. Refer to the previous exercise. Find the mean and median number of drinks for males. Which one is a better representation of how much a typical male who sits in the back of the room drinks? Explain.
29. Refer to the previous exercise. Students were also asked if they typically sit in the front, back, or middle of the classroom. Here are the responses to the question about alcohol consumption for the students who responded that they typically sit in the back of the classroom:Males (N 22): 0,
*28. The students surveyed for the data in Example 4 were also asked “How many alcoholic beverages do you consume in a typical week?” Five-number summaries for males’ and females’ responses are Males Females 2 0 0 10 0 2 0 55 0 17.5a. Draw side-by-side skeletal boxplots for the data.b. Are
27. Draw a boxplot illustrating a data set with each of the following features:a. Skewed to the right with no outliers.b. Bell-shaped with the exception of one outlier at the upper end.c. Values uniformly spread across the range of the data.
26. Suppose you had a choice of two professors for a class in which your grade was very important. They both assign scores on a percentage scale (0 to 100). You can have access to three summary measures of the last 200 scores each professor assigned. Of the summary measures discussed in this
*25. Which set of data is more likely to have a bimodal shape: daily New York City temperatures at noon for the summer months or daily New York City temperatures at noon for an entire year? Explain.
24. Find a set of data of interest to you, such as rents from a newspaper or test scores from a class, with at least 12 numbers. Include the data with your answer.a. Create a five-number summary of the data.b. Create a boxplot of the data.c. Describe the data in a paragraph that would be useful to
23. Give one advantage a stemplot has over a histogram and one advantage a histogram has over a stemplot.
22. What is the variance for the Stanford-Binet IQ test?
*21. Would outliers more heavily influence the range or the quartiles? Explain.
20. Three types of pictures were presented in this chapter: stemplots, histograms, and boxplots. Explain the features of a data set for whicha. Stemplots are most usefulb. Histograms are most usefulc. Boxplots are most useful
19. Suppose a set of test scores is approximately bell-shaped, with a mean of 70 and a range of 50. Approximately, what would the minimum and maximum test scores be?
18. Explain the following statement in words that someone with no training in statistics would understand: The heights of adult males in the United States are bell-shaped, with a mean of about 70 inches and a standard deviation of about 3 inches.
17. Give an example of a measurement for which the mode would be more useful than the median or the mean as an indicator of the “typical” value.
16. Construct an example and draw a histogram for a measurement that you think would be bimodal.
*15. Construct an example and draw a histogram for a measurement that you think would be skewed to the right.
14. Construct an example and draw a histogram for a measurement that you think would be bell-shaped.
*13. Suppose an advertisement reported that the mean weight loss after using a certain exercise machine for 2 months was 10 pounds. You investigate further and discover that the median weight loss was 3 pounds.*a. Explain whether it is most likely that the weight losses were skewed to the right,
12. In each of the following cases, would the mean or the median probably be higher, or would they be about equal?a. Salaries in a company employing 100 factory workers and 2 highly paid executives.b. Ages at which residents of a suburban city die, including everything from infant deaths to the
11. Refer to the data on median family income in Table 7.1; a five-number summary is given in Section 7.3, page 133.a. What is the value of the range?b. What is the value of the Interquartile range?c. What values would be outliers, using the definition of an outlier on page 135? Determine if there
10. The data on hours of sleep discussed in Example 5 also included whether each student was male or female. Here are the separate five-number summaries for“hours of sleep” for the two sexes:Males Females 7 7 6 8 6 8 3 16 3 11a. Two males reported sleeping 16 hours and one reported sleeping 12
*9. Refer to the pulse rate data displayed in the stemplots in Figure 7.2.*a. Find the median.*b. Create a five-number summary.
8. Find the mean and standard deviation of the following set of numbers: 10, 20, 25, 30, 40.
7. All the information contained in the five-number summary for a data set is required for constructing a boxplot. What additional information is required?
6. Give an example of a set of more than five numbers that has a five-number summary of 40 30 40 70 10 40 80
5. Give an example of a set of five numbers with a standard deviation of 0.
*4. Give an example for which the median would be more useful than the mean as a measure of center.
3. Create a histogram for the test scores in Exercise 1. Comment on the shape.
2. Refer to the test scores in Exercise 1.a. Create a five-number summary.b. Create a boxplot.
1. At the beginning of this chapter, the following exam scores were listed and a stemplot for them was shown in Figure 7.1: 75, 95, 60, 93, 85, 84, 76, 92, 62, 83, 80, 90, 64, 75, 79, 32, 78, 64, 98, 73, 88, 61, 82, 68, 79, 78, 80, 55.a. Create a stemplot for the test scores using each 10s value
3. Find the journal article in the New England Journal of Medicine on which Case Study 6.5 is based. Evaluate the study using the seven steps on page 108.
2. Choose one of the news stories in the Appendix and the accompanying material on the CD. Evaluate it using the seven steps on page 108. If all of the required information is not available in the news article, locate the journal article or other source of the research. As part of your analysis,
1. Find a news article about a statistical study. Evaluate it using the seven steps on page 108. If all of the required information is not available in the news article, locate the journal article or other source of the research. As part of your analysis, make sure you discuss step 7 with regard to
Pick one of the news stories in the Appendix that describes an observational study and that has one or more journal articles accompanying it on the CD. Explain what was done in the experiment using the terminology and concepts in this chapter. Discuss the extent to which you agree with the
Pick one of the news stories in the Appendix that describes a randomized experiment and that has one or more journal articles accompanying it on the CD.Explain what was done in the experiment using the terminology and concepts in this chapter. Discuss the extent to which you agree with the
Design and carry out a single-blind study using 10 participants. Your goal is to establish whether people write more legibly with their dominant hand. In other words, do right-handed people write more legibly with their right hand and do left-handed people write more legibly with their left hand?
Go to the library or the Internet and locate a journal article that describes an observational study. Explain how it was done using the terminology of this chapter and whether you agree with the conclusions drawn by the authors.
Go to the library or the Internet and locate a journal article that describes a randomized experiment. Explain what was done correctly and incorrectly in the experiment and whether you agree with the conclusions drawn by the authors.
Design an observational study to test something of interest to you. Explain how your design addresses each of the three complications listed in Section 5.5,“Difficulties and Disasters in Observational Studies.”
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