1/9/2018 thinking mathematically, sixth edition...in exercises 27–30, use table 12.17 on page 816...

1/9/2018 Thinking Mathematically, Sixth Edition

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12 Statistics > 12.6 Scatter Plots, Correlation, and Regression Lines

The weights for 12-month-old baby boys are normally distributed with a mean of 22.5 pounds and a standard deviationof 2.2 pounds. In Exercises 27–30, use Table 12.17 on page 816 to find the percentage of 12-month-old baby boys whoweigh

27. more than 25.8 pounds.

28. more than 23.6 pounds.

29. between 19.2 and 21.4 pounds.

30. between 18.1 and 19.2 pounds.

Practice PlusThe table shows selected ages of licensed drivers in the United States and the corresponding percentiles.

AGES OF U.S.DRIVERS

Age Percentile75 9865 8855 7745 6035 3725 1420 5

Source: Department of Transportation

In Exercises 31–36, use the information given by the table to find the percentage of U.S. drivers who are

31. younger than 55.

32. younger than 45.

33. at least 25.

34. at least 35.

35. at least 65 and younger than 75.

36. at least 20 and younger than 65.

Writing in Mathematics37. Explain when it is necessary to use a table showing z-scores and percentiles rather than the 68–95–99.7 Rule todetermine the percentage of data items less than a given data item.

38. Explain how to use a table showing z-scores and percentiles to determine the percentage of data items betweentwo z-scores.

Critical Thinking ExercisesMake Sense? In Exercises 39–42, determine whether each statement makes sense or does not make sense, andexplain your reasoning.

39. I'm using a table showing z-scores and percentiles that has positive percentiles corresponding to positive z-scoresand negative percentiles corresponding to negative z-scores.

40. My table showing z-scores and percentiles displays the percentage of data items less than a given value of z.

41. My table showing z-scores and percentiles does not display the percentage of data items greater than a givenvalue of z.

42. I can use a table showing z-scores and percentiles to verify the three approximate numbers given by the 68–95–99.7 Rule.

43. Find two z-scores so that 40% of the data in the distribution lies between them. (More than one answer ispossible.)

44. A woman insists that she will never marry a man as short or shorter than she, knowing that only one man in 400falls into this category. Assuming a mean height of 69 inches for men with a standard deviation of 2.5 inches (and anormal distribution), approximately how tall is the woman?

45. The placement test for a college has scores that are normally distributed with a mean of 500 and a standarddeviation of 100. If the college accepts only the top 10% of examinees, what is the cutoff score on the test foradmission?

12.6 Scatter Plots, Correlation, and Regression Lines

What am I Supposed to Learn?

Table of Contents

Skip Directly to Table of Contents | Skip Directly to Main Content

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Thinking Mathematically, SixthEdition

1 Problem Solving and CriticalThinking

2 Set Theory

3 Logic

4 Number Representation andCalculation

5 Number Theory and the RealNumber System

6 Algebra: Equations andInequalities

7 Algebra: Graphs, Functionsand Linear Systems

8 Personal Finance

9 Measurement

10 Geometry

11 Counting Methods andProbability Theory

12 Statistics

12.1 Sampling, FrequencyDistributions, and Graphs

12.2 Measures of CentralTendency

12.3 Measures of Dispersion

12.4 The Normal Distribution

12.5 Problem Solving with theNormal Distribution

12.6 Scatter Plots, Correlation,and Regression Lines

Chapter Summary, Review, andTest

Chapter 12 Test

13 Voting and Apportionment

14 Graph Theory

Answers to Selected Exercises

Credits

Subject Index

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Page 821

After you have read this section, you should be able to:

1 Make a scatter plot for a table of data items.

2 Interpret information given in a scatter plot.

3 Compute the correlation coefficient.

4 Write the equation of the regression line.

5 Use a sample's correlation coefficient to determine whether there is a correlation in the population.

THESE PHOTOS OF PRESIDENTIAL PUFFING INDICATE that the White House was not always a no-smoking zone.According to Cigar Aficionado, nearly half of U.S. presidents have had a nicotine habit, from cigarettes to pipes to cigars.Franklin Roosevelt's stylish way with a cigarette holder was part of his mystique. Although Dwight Eisenhower quit hiswartime four-pack-a-day habit before taking office, smoking in the residence was still common, with ashtrays on thetables at state dinners and free cigarettes for guests. In 1993, Hillary Clinton banned smoking in the White House,although Bill Clinton's cigars later made a sordid cameo in the Lewinsky scandal. Barack Obama quit smoking beforeentering the White House, but had “fallen off the wagon occasionally” as he admitted in a Meet the Press interview.

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Page 822

12 Statistics > 12.6 Scatter Plots, Correlation, and Regression Lines > Scatter Plots and Correlation

Changing attitudes toward smoking, both inside and outside the White House, date back to 1964 and an equation in two variables. To understand the mathematicsbehind this turning point in public health, we need to explore situations involving data collected on two variables.

Up to this point in the chapter, we have studied situations in which data sets involve a single variable, such as height, weight, cholesterol level, and length ofpregnancies. By contrast, the 1964 study involved data collected on two variables from 11 countries—annual cigarette consumption for each adult male and deaths permillion males from lung cancer. In this section, we consider situations in which there are two data items for each randomly selected person or thing. Our interest is indetermining whether or not there is a relationship between the two variables and, if so, the strength of that relationship.

Scatter Plots and Correlation

1 Make a scatter plot for a table of data items.

Is there a relationship between education and prejudice? With increased education, does a person's level of prejudice tend to decrease? Notice that we are interested intwo quantities—years of education and level of prejudice. For each person in our sample, we will record the number of years of school completed and the score on a testmeasuring prejudice. Higher scores on this 1-to-10 test indicate greater prejudice. Using x to represent years of education and y to represent scores on a test measuringprejudice, Table 12.18 shows these two quantities for a random sample of ten people.

TABLE 12.18 Recording Two Quantities in a Sample of TenPeople

Respondent A B C D E F G H I JYears of education (x) 12 5 14 13 8 10 16 11 12 4Score on prejudice test (y) 1 7 2 3 5 4 1 2 3 10

When two data items are collected for every person or object in a sample, the data items can be visually displayed using a scatter plot. A scatter plot is a collection ofdata points, one data point per person or object. We can make a scatter plot of the data in Table 12.18 by drawing a horizontal axis to represent years of education anda vertical axis to represent scores on a test measuring prejudice. We then represent each of the ten respondents with a single point on the graph. For example, the dotfor respondent A is located to represent 12 years of education on the horizontal axis and 1 on the prejudice test on the vertical axis. Plotting each of the ten pieces ofdata in a rectangular coordinate system results in the scatter plot shown in Figure 12.27.

dFIGURE 12.27 A scatter plot for education-prejudice data

A scatter plot like the one in Figure 12.27 can be used to determine whether two quantities are related. If there is a clear relationship, the quantities are said to becorrelated. The scatter plot shows a downward trend among the data points, although there are a few exceptions. People with increased education tend to have a lowerscore on the test measuring prejudice. Correlation is used to determine if there is a relationship between two variables and, if so, the strength and direction of thatrelationship.

Correlation and Causal ConnectionsCorrelations can often be seen when data items are displayed on a scatter plot. Although the scatter plot in Figure 12.27 indicates a correlation between education andprejudice, we cannot conclude that increased education causes a person's level of prejudice to decrease. There are at least three possible explanations:
















12 Statistics > 12.6 Scatter Plots, Correlation, and Regression Lines > Regression Lines and Correlation Coefficients

1. The correlation between increased education and decreased prejudice is simply a coincidence.

2. Education usually involves classrooms with a variety of different kinds of people. Increased exposure to diversity in the classroom setting, which accompaniesincreased levels of education, might be an underlying cause for decreased prejudice.

This list represents three possibilities. Perhaps you can provide a better explanation about decreasing prejudice with increased education.

3. Education, the process of acquiring knowledge, requires people to look at new ideas and see things in different ways. Thus, education causes one to be moretolerant and less prejudiced.

Establishing that one thing causes another is extremely difficult, even if there is a strong correlation between these things. For example, as the air temperatureincreases, there is an increase in the number of people stung by jellyfish at the beach. This does not mean that an increase in air temperature causes more people to bestung. It might mean that because it is hotter, more people go into the water. With an increased number of swimmers, more people are likely to be stung. In short,correlation is not necessarily causation.

Regression Lines and Correlation Coefficients

2 Interpret information given in a scatter plot.

Figure 12.28 shows the scatter plot for the education-prejudice data. Also shown is a straight line that seems to approximately “fit” the data points. Most of the datapoints lie either near or on this line. A line that best fits the data points in a scatter plot is called a regression line. The regression line is the particular line in which thespread of the data points around it is as small as possible.

dFIGURE 12.28 A scatter plot with a regression line

A measure that is used to describe the strength and direction of a relationship between variables whose data points lie on or near a line is called the correlationcoefficient, designated by r. Figure 12.29 shows scatter plots and correlation coefficients. Variables are positively correlated if they tend to increase or decreasetogether, as in Figure 12.29(a), (b), and (c). By contrast, variables are negatively correlated if one variable tends to decrease while the other increases, as in Figure12.29(e), (f), and (g). Figure 12.29 illustrates that a correlation coefficient, r, is a number between and 1, inclusive. Figure 12.29(a) shows a value of 1. Thisindicates a perfect positive correlation in which all points in the scatter plot lie precisely on the regression line that rises from left to right. Figure 12.29(g) shows avalue of This indicates a perfect negative correlation in which all points in the scatter plot lie precisely on the regression line that falls from left to right.



−1

−1.








Page 823

dFIGURE 12.29 Scatter plots and correlation coefficients

Take another look at Figure 12.29. If r is between 0 and 1, as in (b) and (c), the two variables are positively correlated, but not perfectly. Although all the data points willnot lie on the regression line, as in (a), an increase in one variable tends to be accompanied by an increase in the other. Negative correlations are also illustrated inFigure 12.29. If r is between 0 and as in (e) and (f), the two variables are negatively correlated, but not perfectly. Although all the data points will not lie on theregression line, as in (g), an increase in one variable tends to be accompanied by a decrease in the other.


−1,








12 Statistics > 12.6 Scatter Plots, Correlation, and Regression Lines > How to Obtain the Correlation Coefficient and the Equationof the Regression Line

Blitzer BonusBeneficial Uses of Correlation Coefficients

• A Florida study showed a high positive correlation between the number of powerboats and the number of manatee deaths. Many of these deaths were seen to becaused by boats' propellers gashing into the manatees' bodies. Based on this study, Florida set up coastal sanctuaries where powerboats are prohibited so thatthese large gentle mammals that float just below the water's surface could thrive.

• Researchers studied how psychiatric patients readjusted to their community after their release from a mental hospital. A moderate positive correlation was found between patients' attractiveness and their postdischarge social adjustment. The better-looking patients were better off. The researchers suggested thatphysical attractiveness plays a role in patients' readjustment to community living because good-looking people tend to be treated better by others than homelypeople are.

Example 1 Interpreting a Correlation CoefficientIn a 1971 study involving 232 subjects, researchers found a relationship between the subjects' level of stress and how often they became ill. The correlation coefficientin this study was 0.32. Does this indicate a strong relationship between stress and illness?

SOLUTION

The correlation coefficient means that as stress increases, frequency of illness also tends to increase. However, 0.32 is only a moderate correlation,illustrated in Figure 12.29(c) on the previous page. There is not, based on this study, a strong relationship between stress and illness. In this study, the relationship issomewhat weak.

Check Point 1In a 1996 study involving obesity in mothers and daughters, researchers found a relationship between a high body-mass index for the girls and their mothers. (Body-mass index is a measure of weight relative to height. People with a high body-mass index are overweight or obese.) The correlation coefficient in this study was 0.51.Does this indicate a weak relationship between the body-mass index of daughters and the body-mass index of their mothers?

How to Obtain the Correlation Coefficient and the Equation of the Regression LineThe easiest way to find the correlation coefficient and the equation of the regression line is to use a graphing or statistical calculator. Graphing calculators have statisticalmenus that enable you to enter the x and y data items for the variables. Based on this information, you can instruct the calculator to display a scatter plot, the equation ofthe regression line, and the correlation coefficient.

We can also compute the correlation coefficient and the equation of the regression line by hand using formulas. First, we compute the correlation coefficient.

Computing the Correlation Coefficient by HandThe following formula is used to calculate the correlation coefficient, r:

In the formula,



(r = 0.38)

r = 0.32

r = .n (∑xy) − (∑x) (∑y)

n (∑ ) −x2 (∑x)2− −−−−−−−−−−−−−−

√ n (∑ ) −y2 (∑y)2− −−−−−−−−−−−−−

√








Page 824


n

∑x

∑y

∑xy

∑x2

∑y2

(∑x)2

(∑y)2

=

=

=

=

=

=

=

=

the number of data points, (x, y)

the sum of the x-values

the sum of the y-values

the sum of the product of x and y in each pair

the sum of the squares of the x-values

the sum of the squares of the y-values

the square of the sum of the x-values

the square of the sum of the y-values









When computing the correlation coefficient by hand, organize your work in five columns:

x y xyFind the sum of the numbers in each column. Then, substitute these values into the formula for r. Example 2 illustrates computing the correlation coefficient for theeducation-prejudice test data.

TechnologyGraphing Calculators, Scatter Plots, and Regression Lines

You can use a graphing calculator to display a scatter plot and the regression line. After entering the x and y data items for years of education and scores on aprejudice test, the calculator shows the scatter plot of the data and the regression line.

d

Also displayed below is the regression line's equation and the correlation coefficient, r. The slope shown below is approximately The negative slopereinforces the fact that there is a negative correlation between the variables in Example 2.

d

Example 2 Computing the Correlation Coefficient

3 Compute the correlation coefficient.

Shown below are the data involving the number of years of school, x, completed by ten randomly selected people and their scores on a test measuring prejudice, y.Recall that higher scores on the measure of prejudice (1 to 10) indicate greater levels of prejudice. Determine the correlation coefficient between years of educationand scores on a prejudice test.

Respondent A B C D E F G H I JYears of education (x) 12 5 14 13 8 10 16 11 12 4Score on prejudice test (y) 1 7 2 3 5 4 1 2 3 10

SOLUTION

As suggested, organize the work in five columns.

x y xy12 1 12 144 15 7 35 25 4914 2 28 196 413 3 39 169 98 5 40 64 2510 4 40 100 1616 1 16 256 111 2 22 121 412 3 36 144 9



x2 y2

−0.69.

x2 y2

4 10 40 16 100








Page 825

x y xy

We use these five sums to calculate the correlation coefficient.

Another value in the formula for r that we have not yet determined is n, the number of data points (x, y). Because there are ten items in the x-column and ten items inthe y-column, the number of data points (x, y) is ten. Thus,

In order to calculate r, we also need to find the square of the sum of the x-values and the y-values:


∑x = 105

▲

Add all values in

the x-column.

∑y = 38

▲

Add all values in

the y-column.

∑xy = 308

▲

Add all values in

the xy-column.

∑ = 1235x2

▲

Add all values in

the -column.x2

∑ = 218y2

▲

Add all values in

the -column.y2

n = 10.

= = 11,025 and = = 1444.(∑x)2

(105)2 (∑y)2

(38)2








Page 826


We are ready to determine the value for r. We use the sums obtained on the previous page, with

The value for r, approximately is fairly close to and indicates a strong negative correlation. This means that the more education a person has, the lessprejudiced that person is (based on scores on the test measuring levels of prejudice).

Check Point 2The points in the scatter plot in Figure 12.30 show the number of firearms per 100 persons and the number of deaths per 100,000 persons for the ten industrializedcountries with the highest death rates. Use the data displayed by the voice balloons to determine the correlation coefficient between these variables. Round to twodecimal places. What does the correlation coefficient indicate about the strength and direction of the relationship between firearms per 100 persons and deaths per100,000 persons?

dFIGURE 12.30 Source: International Action Network on Small Arms

Once we have determined that two variables are related, we can use the equation of the regression line to determine the exact relationship. Here is the formula forwriting the equation of the line that best fits the data:

4 Write the equation of the regression line.

Writing the Equation of the Regression Line by HandThe equation of the regression line is

where



n = 10

r

=

=

=

≈

n(∑ xy)−(∑ x)(∑ y)

n(∑ )−x2 (∑ x)2√ n(∑ )−y2 (∑ y)2√10(308)−105(38)

10(1235)−11,025√ 10(218)−1444√−910

1325√ 736√

−0.92

−0.92, −1

y = mx + b,

m = and b = .n (∑xy) − (∑x) (∑y)

n (∑ ) −x2 (∑x)2

∑y − m (∑x)

n








12 Statistics > 12.6 Scatter Plots, Correlation, and Regression Lines > The Level of Significance of r

Example 3 Writing the Equation of the Regression Linea. Shown, again, in Figure 12.28 is the scatter plot and the regression line for the data in Example 2. Use the data to find the equation of the regression line thatrelates years of education and scores on a prejudice test.

b. Approximately what score on the test can be anticipated by a person with nine years of education?

FIGURE 12.28 (repeated)

SOLUTION

a. We use the sums obtained in Example 2. We begin by computing m.

With a negative correlation coefficient, it makes sense that the slope of the regression line is negative. This line falls from left to right, indicating a negativecorrelation.

Now, we find the y-intercept, b.

Using and the equation of the regression line, is

where x represents the number of years of education and y represents the score on the prejudice test.

b. To anticipate the score on the prejudice test for a person with nine years of education, substitute 9 for x in the regression line's equation.

A person with nine years of education is anticipated to have a score close to 5 on the prejudice test.

Great Question!Why is in Example 3, but in the Technology box on page 825?

In Example 3, we rounded the value of m when we calculated b. The value of b on the calculator screen on page 825 is more accurate.

Check Point 3Use the data in Figure 12.30 of Check Point 2 on page 826 to find the equation of the regression line. Round m and b to one decimal place. Then use the equation toproject the number of deaths per 100,000 persons in a country with 80 firearms per 100 persons.

The Level of Significance of r

5 Use a sample's correlation coefficient to determine whether there is a correlation in the population.



m = = = ≈ −0.69n (∑xy) − (∑x) (∑y)

n (∑ ) −x2 (∑x)2

10 (308) − 105 (38)

10 (1235) − (105)2

−910

1325

b = = = ≈ 11.05∑y − m (∑x)

n

38 − (−0.69) (105)

10

110.45

10

m ≈ −0.69 b ≈ 11.05, y = mx + b,

y = −0.69x + 11.05,

y

y

=

=

−0.69x + 11.05

−0.69 (9) + 11.05 = 4.84

b ≈ 11.05 b ≈ 11.01











Page 827

In Example 2, we found a strong negative correlation between education and prejudice, computing the correlation coefficient, r, to be However, the sample size was relatively small. With such a small sample, can we truly conclude that a correlation exists in the population? Or could it be that education and prejudice

are not related? Perhaps the results we obtained were simply due to sampling error and chance.

Mathematicians have identified values to determine whether r, the correlation coefficient for a sample, can be attributed to a relationship between variables in thepopulation.


−0.92.(n = 10)








12 Statistics > 12.6 Scatter Plots, Correlation, and Regression Lines > The Level of Significance of r

These values are shown in the second and third columns of Table 12.19. They depend on the sample size, n, listed in the left column. If the absolute value of thecorrelation coefficient computed for the sample, is greater than the value given in the table, a correlation exists between the variables in the population. The columnheaded denotes a significance level of 5%, meaning that there is a 0.05 probability that, when the statistician says the variables are correlated, they areactually not related in the population. The column on the right, headed denotes a significance level of 1%, meaning that there is a 0.01 probability that,when the statistician says the variables are correlated, they are actually not related in the population. Values in the column are greater than those in the

column. Because of the possibility of sampling error, there is always a probability that when we say the variables are related, there is actually not acorrelation in the population from which the sample was randomly selected.

TABLE 12.19 Values forDetermining Correlations in a

Populationn

4 0.950 0.9905 0.878 0.9596 0.811 0.9177 0.754 0.8758 0.707 0.8349 0.666 0.79810 0.632 0.76511 0.602 0.73512 0.576 0.70813 0.553 0.68414 0.532 0.66115 0.514 0.64116 0.497 0.62317 0.482 0.60618 0.468 0.59019 0.456 0.57520 0.444 0.56122 0.423 0.53727 0.381 0.48732 0.349 0.44937 0.325 0.41842 0.304 0.39347 0.288 0.37252 0.273 0.35462 0.250 0.32572 0.232 0.30282 0.217 0.28392 0.205 0.267102 0.195 0.254

The larger the sample size, n, the smaller is the value of r needed for a correlation in the population.

Example 4 Determining a Correlation in the PopulationIn Example 2, we computed for Can we conclude that there is a negative correlation between education and prejudice in the population?

SOLUTION

Begin by taking the absolute value of the calculated correlation coefficient.

Now, look to the right of in Table 12.19. Because 0.92 is greater than both of these values (0.632 and 0.765), we may conclude that a correlation does existbetween education and prejudice in the population. (There is a probability of at most 0.01 that the variables are not really correlated in the population and our resultscould be attributed to chance.)

Blitzer BonusCigarettes and Lung Cancer



|r| ,

α = 0.05α = 0.01,

α = 0.01α = 0.05

α = 0.05 α = 0.01

r = −0.92 n = 10.

|r| = |−0.92| = 0.92

n = 10








Page 828

dSource: Smoking and Health, Washington, D.C., 1964

This scatter plot shows a relationship between cigarette consumption among males and deaths due to lung cancer per million males. The data are from 11 countriesand date back to a 1964 report by the U.S. Surgeon General. The scatter plot can be modeled by a line whose slope indicates an increasing death rate from lungcancer with increased cigarette consumption. At that time, the tobacco industry argued that in spite of this regression line, tobacco use is not the cause of cancer.Recent data do, indeed, show a causal effect between tobacco use and numerous diseases.

Check Point 4If you worked Check Point 2 correctly, you should have found that for Can you conclude that there is a positive correlation for all industrializedcountries between firearms per 100 persons and deaths per 100,000 persons?


r ≈ 0.89 n = 10.








12 Statistics > 12.6 Scatter Plots, Correlation, and Regression Lines > Concept and Vocabulary Check

Concept and Vocabulary CheckFill in each blank so that the resulting statement is true.

1. A set of points representing data is called a/an ______________________.

2. The line that best fits a set of points is called a/an ______________________.

3. A measure that is used to describe the strength and direction of a relationship between variables whose data points lie on or near a line is called the___________________________, ranging from to

In Exercises 4–7, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

4. If there is no correlation between two variables. ________________

5. If changes in one variable cause changes in the other variable. ________________

6. If there is a strong negative correlation between two variables. ________________

7. A significance level of 5% means that there is a 0.05 probability that when a statistician says that variables are correlated, they are actually not related in thepopulation. ________________

Exercise Set 12.6Practice and Application ExercisesIn Exercises 1–8, make a scatter plot for the given data. Use the scatter plot to describe whether or not the variables appear to be related.

1.

x 1 6 4 3 7 2y 2 5 3 3 4 1

2.

x 2 1 6 3 4y 4 5 10 8 9

3.

x 8 6 1 5 4 10 3y 2 4 10 5 6 2 9

4.

x 4 5 2 1y 1 3 5 4

5.

HAMACHIPHOBIA

GenerationPercentage WhoWon't Try Sushi x

Don't Approve of Marriage Equality y

Millennials 42 36Gen X 52 49Boomers 60 59Silent/Greatest Generation 72 66

Source: Pew Research Center

6.

TREASURED CHEST: FILMS OF MATTHEW MCCONAUGHEY

Film Minutes Shirtless x

Opening Weekend Gross (millions of dollars) y

We Are Marshall 0 6.1EDtv 0.8 8.3Reign of Fire 1.6 15.6Sahara 1.8 18.1Fool's Gold 14.6 21.6



r =_________ r =_________ .

r = 0,

r = 1,

r = −0.1,








Page 829

Film Minutes Shirtless x

Opening Weekend Gross (millions of dollars) y

Source: Entertainment Weekly

7.

TEENAGE DRUG USE

CountryPercentage Who Have UsedMarijuana x

Other Illegal Drugs y

Czech Republic 22 4Denmark 17 3England 40 21Finland 5 1Ireland 37 16Italy 19 8Northern Ireland 23 14Norway 6 3Portugal 7 3Scotland 53 31United States 34 24

Source: De Veaux et.al., Intro Stats, Pearson, 2009.










8.

LITERACY AND HUNGER

CountryPercentage Who AreLiterate x

Undernourished y

Cuba 100 2Egypt 71 4Ethiopia 36 46Grenada 96 7Italy 98 2Jamaica 80 9Jordan 91 6Pakistan 50 24Russia 99 3Togo 53 24Uganda 67 19

Source: The Penguin State of the World Atlas, 2008

The scatter plot in the figure shows the relationship between the percentage of married women of child-bearing age using contraceptives and births per woman inselected countries. Use the scatter plot to determine whether each of the statements in Exercises 9–18 is true or false.

dSource: Population Reference Bureau

9. There is a strong positive correlation between contraceptive use and births per woman.

10. There is no correlation between contraceptive use and births per woman.

11. There is a strong negative correlation between contraceptive use and births per woman.

12. There is a causal relationship between contraceptive use and births per woman.

13. With approximately 43% of women of child-bearing age using contraceptives, there are three births per woman in Chile.

14. With 20% of women of child-bearing age using contraceptives, there are six births per woman in Vietnam.

15. No two countries have a different number of births per woman with the same percentage of married women using contraceptives.

16. The country with the greatest number of births per woman also has the smallest percentage of women using contraceptives.

17. Most of the data points do not lie on the regression line.

18. The number of selected countries shown in the scatter plot is approximately 20.

Just as money doesn't buy happiness for individuals, the two don't necessarily go together for countries either. However, the scatter plot does show a relationshipbetween a country's annual per capita income and the percentage of people in that country who call themselves “happy.” Use the scatter plot to determine whether eachof the statements in Exercises 19–26 is true or false.










dSource: Richard Layard, Happiness: Lessons from a New Science, Penguin, 2005

19. There is no correlation between per capita income and the percentage of people who call themselves “happy.”

20. There is an almost-perfect positive correlation between per capita income and the percentage of people who call themselves “happy.”

21. There is a positive correlation between per capita income and the percentage of people who call themselves “happy.”

22. As per capita income decreases, the percentage of people who call themselves “happy” also tends to decrease.

23. The country with the lowest per capita income has the least percentage of people who call themselves “happy.”

24. The country with the highest per capita income has the greatest percentage of people who call themselves “happy.”

25. A reasonable estimate of the correlation coefficient for the data is 0.8.

26. A reasonable estimate of the correlation coefficient for the data is

Use the scatter plots shown, labeled (a)–(f), to solve Exercises 27–30.

(a) d

(b) d

(c) d

(d) d

(e) d

(f) d

−0.3.



Page 830










27. Which scatter plot indicates a perfect negative correlation?

28. Which scatter plot indicates a perfect positive correlation?

29. In which scatter plot is

30. In which scatter plot is

Compute r, the correlation coefficient, rounded to two decimal places, for the data in

31. Exercise 1.

32. Exercise 2.

33. Exercise 3.

34. Exercise 4.

35. Use the data in Exercise 5 to solve this exercise.

a. Determine the correlation coefficient, rounded to two decimal places, between the percentage of people who won't try sushi and the percentage who don'tapprove of marriage equality.

b. What explanations can you offer for the correlation coefficient in part (a)?

c. Find the equation of the regression line for the percentage who won't try sushi and the percentage who don't approve of marriage equality. Round m and b to twodecimal places.

d. What percentage of people, to the nearest percent, can we anticipate do not approve of marriage equality in a generation where 30% won't try sushi?


a. Determine the correlation coefficient, rounded to two decimal places, between the minutes Matthew McConaughey appeared shirtless in a film and the film'sopening weekend gross.

b. Find the equation of the regression line for the minutes McConaughey appeared shirtless in a film and the film's opening weekend gross. Round m and b to twodecimal places.

c. What opening weekend gross, to the nearest tenth of a million dollars, can we anticipate in a McConaughey film in which he appears shirtless for 20 minutes?


a. Determine the correlation coefficient, rounded to two decimal places, between the percentage of teenagers who have used marijuana and the percentage whohave used other drugs.

b. Find the equation of the regression line for the percentage of teenagers who have used marijuana and the percentage who have used other drugs. Round m andb to two decimal places.

c. What percentage of teenagers, to the nearest percent, can we anticipate using illegal drugs other than marijuana in a country where 10% of teenagers have usedmarijuana?


a. Determine the correlation coefficient, rounded to two decimal places, between the percentage of people in a country who are literate and the percentage who areundernourished.

b. Find the equation of the regression line for the percentage who are literate and the percentage who are undernourished. Round m and b to two decimal places.

c. What percentage of people, to the nearest percent, can we anticipate are undernourished in a country where 60% of the people are literate?

In Exercises 39–45, the correlation coefficient, r, is given for a sample of n data points. Use the column in Table 12.19 on page 828 to determine whether ornot we may conclude that a correlation does exist in the population. (Using the column, there is a probability of 0.05 that the variables are not reallycorrelated in the population and our results could be attributed to chance. Ignore this possibility when concluding whether or not there is a correlation in the population.)

39.

40.

41.

42.

43.

44.

45.

46. In the 1964 study on cigarette consumption and deaths due to lung cancer (see the Blitzer Bonus on page 828), and What can you concludeusing the column in Table 12.19 on page 828?



r = 0.9?

r = 0.01?

α = 0.05α = 0.05

n = 20, r = 0.5

n = 27, r = 0.4

n = 12, r = 0.5

n = 22, r = 0.04

n = 72, r = −0.351

n = 37, r = −0.37

n = 20, r = −0.37

n = 11 r = 0.73.α = 0.05











Page 831

Writing in Mathematics47. What is a scatter plot?

48. How does a scatter plot indicate that two variables are correlated?

49. Give an example of two variables with a strong positive correlation and explain why this is so.

50. Give an example of two variables with a strong negative correlation and explain why this is so.

51. What is meant by a regression line?

52. When all points in a scatter plot fall on the regression line, what is the value of the correlation coefficient? Describe what this means.

For the pairs of quantities in Exercises 53–56, describe whether a scatter plot will show a positive correlation, a negative correlation, or no correlation. If there is acorrelation, is it strong, moderate, or weak? Explain your answers.

53. Height and weight

54. Number of days absent and grade in a course

55. Height and grade in a course

56. Hours of television watched and grade in a course

57. Explain how to use the correlation coefficient for a sample to determine if there is a correlation in the population.

Critical Thinking ExercisesMake Sense? In Exercises 58–61, determine whether each statement makes sense or does not make sense, and explain your reasoning.

58. I found a strong positive correlation for the data in Exercise 7 relating the percentage of teenagers in various countries who have used marijuana and thepercentage who have used other drugs. I concluded that using marijuana leads to the use of other drugs.

59. I found a strong negative correlation for the data in Exercise 8 relating the percentage of people in various countries who are literate and the percentage who areundernourished. I concluded that an increase in literacy causes a decrease in undernourishment.









Page 832

12 Statistics > Chapter Summary, Review, and Test

60. I'm working with a data set for which the correlation coefficient and the slope of the regression line have opposite signs.

61. I read that there is a correlation of 0.72 between IQ scores of identical twins reared apart, so I would expect a significantly lower correlation, approximately 0.52,between IQ scores of identical twins reared together.

62. Give an example of two variables with a strong correlation, where each variable is not the cause of the other.

Technology Exercise63. Use the linear regression feature of a graphing calculator to verify your work in any two exercises from Exercises 35–38, parts (a) and (b).

Group Exercises64. The group should select two variables related to people on your campus that it believes have a strong positive or negative correlation. Once these variables havebeen determined,

a. Collect at least 30 ordered pairs of data (x, y) from a sample of people on your campus.

b. Draw a scatter plot for the data collected.

c. Does the scatter plot indicate a positive correlation, a negative correlation, or no relationship between the variables?

d. Calculate r. Does the value of r reinforce the impression conveyed by the scatter plot?

e. Find the equation of the regression line.

f. Use the regression line's equation to make a prediction about a y-value given an x-value.

g. Are the results of this project consistent with the group's original belief about the correlation between the variables, or are there some surprises in the datacollected?

65. What is the opinion of students on your campus about …? Group members should begin by deciding on some aspect of college life around which student opinioncan be polled. The poll should consist of the question, “What is your opinion of …?” Be sure to provide options such as excellent, good, average, poor, horrible, or a 1-to-10 scale, or possibly grades of A, B, C, D, F. Use a random sample of students on your campus and conduct the opinion survey. After collecting the data, presentand interpret it using as many of the skills and techniques learned in this chapter as possible.

Chapter Summary, Review, and TestSUMMARY – DEFINITIONS AND CONCEPTS EXAMPLES

12.1 Sampling, Frequency Distributions, and Graphsa. A population is the set containing all objects whose properties are to be described and analyzed. A sample is a subset of the population. Ex. 1, p. 767b. Random samples are obtained in such a way that each member of the population has an equal chance of being selected. Ex. 2, p. 768

c. Data can be organized and presented in frequency distributions, grouped frequency distributions, histograms, frequency polygons, andstem-and-leaf plots.

Ex. 3, p. 770 ; Ex. 4, p. 771 ; Figures 12.2 and 12.3, p.772 ; Ex. 5, p. 772

d. The box on page 774 lists some things to watch for in visual displays of data. Table 12.5, p. 77512.2 Measures of Central Tendency

a. The mean, is the sum of the data items divided by the number of items: Ex. 1, p. 780

b. The mean, of a frequency distribution is computed using

where x is a data value, f is its frequency, and n is the total frequency of the distribution.

Ex. 2, p. 782

c. The median of ranked data is the item in the middle or the mean of the two middlemost items. The median is the value in the position in the list of ranked data.

Ex. 3, p. 783 ; Ex. 4, p. 784 ; Ex. 5, p. 785 ; Ex. 6, p. 786




,x̄ = .x̄∑ x

n

,x̄

= ,x̄∑xf

n

n+1

2
























1/9/2018 thinking mathematically, sixth edition...in exercises 27–30, use table 12.17 on page 816...

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