Statistical Inference for Bivariate Populations

Abstract

Many questions of interest (both in research and in applications) involve the relationship between two simultaneously collected variables (bivariate observations). For example, is there a relationship between the size of alumni donations to the general fund of a university and the performance of its basketball and football teams? How does the amount of annual rainfall affect the wheat yield in the United States? Does the amount of fracking wastewater injected into deep wells have an effect on the number and severity of earthquakes in the region? Is there any relationship between CO₂ production and sea levels? How does a prescribed diet-medication regimen affect blood pressure levels in subjects with severe high blood pressures? Is there any relationship between pine needle length and diameter of a pine tree? Does smoking or excessive drinking have an impact on mortality? Problems such as these are addressed statistically through the use of correlation or regression analyses.

Chapter 11 Comprehensive Exercises

11.1.1 11.A. Conceptual

11.A.1. Let R ₁, …, R _n be the ranks (from least to greatest) of the variables X ₁, …, X _n, respectively.

• (a) Use algebra to show that $\overline{R}=\frac{1}{n}\sum_{i=1}^n{R}_i=\frac{n+1}{2}$.

• (b) Use algebra to show that $\sum_{i=1}^n{\left({R}_i-\overline{R}\right)}^2=\frac{n\left({n}^2-1\right)}{2}$.

11.A.2. Use algebra to show that the two expressions for the slope estimator in (11.11) are equivalent. That is, show that

$\frac{\sum_{i=1}^n\left({x}_i-\overline{x}\right)\left({y}_i-\overline{y}\right)}{\sum_{j=1}^n{\left({x}_j-\overline{x}\right)}^2}=\frac{n\sum_{i=1}^n{x}_i{y}_i-\left(\sum_{j=1}^n{x}_j\right)\left(\sum_{k=1}^n{y}_k\right)}{n\sum_{i=1}^n{x}_i^2-{\left(\sum_{j=1}^n{x}_j\right)}^2}$.

11.A.3. Use algebra to show that the value of the Spearman sample correlation coefficient R _S (11.6) can be obtained by using the computationally simpler expression

$$ {R}_S=1-\frac{6\;\sum_{i=1}^n{D}_i^2}{n\left({n}^2-1\right)}, $$

where D _i  = S _i  − R _i , for i = 1,…, n.

11.A.4. Construct a set of bivariate observations (X ₁, Y ₁), …, (X _n, Y _n) for which the Pearson sample correlation coefficient R (11.1) is 0 (least indicative of a linear relationship between X and Y) but for which Y can be expressed as an explicit function of X (so their true relationship is perfect).

11.A.5. Construct a set of bivariate observations (X ₁, Y ₁), …, (X _n, Y _n) for which the Pearson correlation coefficient R (11.1) is 0 (least indicative of a linear relationship between X and Y) but for which the Spearman rank correlation coefficient R _S (11.6) and the Kendall correlation coefficient K (11.18) are both 1 (most indicative of a monotone positive relationship between X and Y).

11.A.6. Let (X, Y) be bivariate random variables. The population correlation between X and Y is defined by

$$ {\rho}_{X,Y}=\frac{E\left[\left(X-{\mu}_X\right)\left(Y-{\mu}_Y\right)\right]}{\sigma_X{\sigma}_Y}, $$

where μ _X  = E(X), μ _Y  = E(Y), ${\sigma}_X^2= Var(X)$, and ${\sigma}_Y^2= Var(Y)$.

• (a) Show that E[(X − μ _X )(Y − μ _Y )] = E[XY] − μ _X μ _Y .

• (b) Show that ρ _X,Y = 0 if X and Y are independent random variables.

• (c) Is the converse to the statement in part (b) also true? That is, does ρ _{X, Y} = 0 also imply that X and Y are independent random variables? Justify your answer.

  [Hint: Consider the joint probability distribution given by

$$ P\left(\left(X,Y\right)=\left(-1,0\right)\right)=P\left(\left(X,Y\right)=\left(0,0\right)\right)=P\left(\left(X,Y\right)=\left(1,0\right)\right)=\frac{1}{3}.\Big] $$

11.A.7. Let X be a positive random variable (i.e., P(X > 0) = 1). Define a second random variable Y = X ². Clearly X and Y are dependent random variables, but they are not linearly related. As a result, the test procedures in (11.3), (11.4), and (11.5) based on the Pearson sample correlation coefficient R will not be particularly effective in detecting this dependence.

• (a) Will the test procedures in (11.8), (11.9), and (11.10) based on the Spearman sample correlation coefficient R _S be capable of detecting this dependence? Justify your answer.

• (b) Compare and contrast the linear regression and monotonic regression procedures discussed in Sects. 4 and 5, respectively, for this setting.

11.A.8. Let (X ₁, Y ₁) ,  …  , (X _n , Y _n ) be a random sample from a bivariate distribution. The Kendall sample correlation coefficient for these data is defined by

$$ K = \frac{\sum_{i=1}^{n-1}\sum_{j=i+1}^nc\left[\left({Y}_j-{Y}_i\right)\left({X}_j-{X}_i\right)\right]}{\left[\frac{n\left(n-1\right)}{2}\right]} $$

where c(t) = −1, 0, 1 if t <, =, > 0. Thus, for each pair of subscripts (i, j), with 1 ≤ i < j ≤ n, score 1 if Y _j > Y _i and X _j > X _i, score − 1 if Y _j < Y _i and X _j > X _i, and score 0 if either X _j = X_i or Y _j = Y _i.

• (a) Construct a data set for which K = −1.

• (b) Construct a data set for which K = 1.

• (c) Construct a data set for which K = 0.

11.A.9. Let R _S be the Spearman sample correlation coefficient defined in (11.6).

• (a) Construct a data set for which R _S  = −1.

• (b) Construct a data set for which R _S  = 1.

• (c) Construct a data set for which R _S  = 0.

11.A.10. Suppose you were interested in testing for a linear or monotonic regression with fixed null hypothesis slope value β ₀ ≠ 0. Discuss how you might use the shifted sample data ${Y}_1 -{\beta}_0{x}_1,\dots, {Y}_{\mathrm{n}}-{\beta}_0{x}_n$ in conjunction with either the linear regression procedures based on T (11.14) or the monotonic regression procedures based on K (11.18) to test the null hypothesis H ₀ :  β  =  β ₀ against appropriate alternatives β > β ₀ ,  β < β ₀, or β ≠ β ₀.

11.A.11. Confidence Interval for the Slope Parameter β for a Normal Population . Let $\widehat{\beta}$ be the slope estimator given in (11.11) and let S _xx  ,  S _xy  ,  and S _yy be as defined in (11.14). When the underlying distribution is bivariate normal, a 100CL% confidence interval for the slope parameter β is then given by

$$ \widehat{\beta}\pm {t}_{n-2,\frac{\left(1,-, CL\right)}{2}}\frac{\sqrt{\frac{S_{yy}-\widehat{\beta}\;{S}_{xy}}{n-2}}}{\sqrt{S_{xx}}}, $$

where ${t}_{n-2,\frac{\left(1,-, CL\right)}{2}}$ is the upper $\frac{\left(1,-, CL\right)}{2}$ percentile for the t-distribution with n – 2 degrees of freedom. The R functions $\text{confint}\left(\right)$ and $\mathit{lm}\left(\right)$ can be used together to obtain this confidence interval for β.

Use the R dataset pines_1997 to obtain a 95% confidence interval for the slope parameter associated with a linear regression of the 1997 height (Hgt97) on the 1997 diameter (Diam97) for the pines at the Kenyon Center for Environmental Study (KCES).

11.A.12. Confidence Interval for the Slope Parameter β for an Arbitrary Population . Consider the linear model setting in (11.13) and let Y _i denote the value of the dependent variable Y at fixed value x _i of the independent variable x, for i = 1, …, n. Suppose that all n x’s are distinct. Compute the N = $\left(\underset{2}{\overset{n}{}}\right)$ = n(n – 1)/2 individual sample slopes

$$ {S}_{ij}=\frac{\left({Y}_j-{Y}_i\right)}{\left({x}_j-{x}_i\right)}, 1\le i<j\le n; $$

and let S ⁽¹⁾ ≤  ⋯  ≤ S ^(N) be the ordered sample slope values. By properly specifying the probs argument, the R function $\text{quantile}\left(\right)$ can be used to obtain a 100CL% confidence interval for β based directly on these ordered sample slopes S ⁽¹⁾ ≤  ⋯  ≤ S ^(N). For example, given a vector of data x, we can construct a 90% confidence interval using the following command.

Use the S _ij sample slope values for the cricket chirp rate data from Table 11.12 in Example 11.5 (without the (x, y) pair (80.6, 17.1) to avoid ties among the temperature values) to find a 95% confidence interval for the slope parameter associated with a linear regression of cricket chirp rate on concurrent temperature.

11.1.2 11.B. Data Analysis/Computational

11.B.1. Do Strikeouts Affect Batting Averages ? In Example 11.1 we found that there was insufficient evidence to link the number of strikeouts with the number of home runs hit by major league ballplayers. A related question of interest is whether the number of strikeouts might be negatively related to the overall batting average for major leaguers. The data in Table 11.17 contains the number of strikeouts and final batting average from the 2016 season for the same 15 major league ballplayers considered in Example 11.1.

Table 11.17 2016 batting average and strikeout statistics for a subset of major league players

Use these data to test the conjecture that a major leaguer’s batting average is negatively correlated with his number of strikeouts.

11.B.2. Do Golf Handicaps “Drive” Stock Prices ? An investment compensation expert, Graef Crystal, undertook a study to investigate whether there is any link between the golf handicap for a company’s CEO and the value of the company’s publicly traded stock. He reported his findings in the May 31, 1998 issue of The New York Times under the heading “Investing It: Duffers Need Not Apply”. Table 11.18 contains the golf handicaps (based on data obtained from the journal Golf Digest) and stock ratings (compiled by Crystal using data on the stock market performance of the companies) for 51 CEO’s.

Table 11.18 Golf handicaps and stock ratings for major company CEOs

Find the P-value for a test of the conjecture that CEO handicap and Stock Rating are negatively correlated.

11.B.3. Voter Turnout in Presidential Elections . The population of the United States has steadily grown over the years since it became a nation, so we might expect that voter turnout in presidential elections would also have grown consistently over time from election to election. Table 11.19 contains the total popular vote (in thousands) for each of the elections from 1940 through 2012.

Table 11.19 Popular vote (in thousands) for presidential elections , 1940–2012

Find the P-value for an appropriate procedure to test if there is, indeed, an increasing trend in popular vote turnout for presidential elections over the period 1940–2012.

11.B.4. Do Caution Flags Really Slow Races? The National Association for Stock Car Auto Racing (NASCAR) was founded in December 1947. It sponsors the Winston Cup, currently comprised of 36 races per year, with up to 43 cars competing in each race. One of the issues surrounding these races is the potential impact on fan enjoyment from slowing the race due to caution flags required when an incident (usually an accident) has occurred. The number of caution flags and winning time (in minutes) for 82 races over the period of time from 1975 through 2003 are given in Table 11.20. Each of these 82 races was held on a 2.5 mile track and the winner of each race completed the full 200 laps. (Thus, the winning times can be compared fairly.)

Table 11.20 Number of caution flags and winning times (in minutes) for 82 NASCAR races during 1975–2003

• (a) Obtain the fitted least squares line for these data, treating number of caution flags as the independent variable and winning time as the dependent variable.

• (b) Find the P-value for a test of the null hypothesis that the winning time for a NASCAR race of 200 laps on a 2.5 mile track is linearly related to the number of caution flags. Which alternative hypothesis do you think is appropriate for this setting?

• (c) Use the fitted least squares line from part (a) to predict the winning time for a race of this type with 6 caution flags. Compare this predicted value with the observed winning times for those races with 6 caution flags in our sample data.

• (d) How would you feel about using the fitted least squares line in (a) to predict winning times for races with no caution flags? races with 15 caution flags?

11.B.5. Come On—I Just Walked Him! There is a general conception in baseball that walks seem to somehow come back to haunt a pitcher by scoring. But are walks really a major contributor to a pitcher’s earned run average? Table 11.21 contains the nine-inning walk rates and the earned run averages for fifteen major leaguers who pitched at least 140 innings during the 2016 baseball season.

Table 11.21 Nine-inning walk rate and earned run average for 15 major leaguers who pitched at least 140 innings in the 2016 season

Use these data to test the conjecture that a major league pitcher’s earned run average is positively correlated with his nine-inning walk rate.

11.B.6. Careful When You Chirp. Consider the cricket chirp rate data from Table 11.12 in Example 11.5.

• (a) What is the equation of the least squares line fit to these data (consider temperature as the independent variable x and cricket chirp rate as the dependent variable y)?

• (b) Using this least squares fitted line, what would you estimate the chirp rate to be when the concurrent temperature is 70 ° F?

• (c) Using this least squares fitted line, what would you estimate the chirp rate to be when the concurrent temperature is 100 ° F?

• (d) Using this least squares fitted line, what would you estimate the chirp rate to be when the concurrent temperature is 32 ° F?

• (e) Discuss your answers to parts (c) and (d) in the context of these data.

11.B.7. How Important Is a Good Starting Position? The National Association for Stock Car Auto Racing (NASCAR) was founded in December 1947. It sponsors the Winston Cup, currently comprised of 36 races per year, with up to 43 cars competing in each race. Is it important to have a good starting position in these races? Table 11.22 contains the starting position and finish position for 42 drivers in a NASCAR race held in 1987.

Table 11.22 Finish position and starting position for each of 42 drivers in a NASCAR race in 1987

• (a) Using the procedure based on the Spearman sample correlation coefficient, find the P-value for a test of the hypothesis that starting position and finish position are independent variables against an appropriate alternative.

• (b) Repeat part (a) using the Kendall sample correlation coefficient. Are your findings similar?

11.B.8. How Important Is a Good Starting Position?—Part II. Table 11.23 contains the starting position and finish position for 42 drivers in a second NASCAR race held in 1997.

Table 11.23 Finish position and starting position for each of 42 drivers in a NASCAR race in 1997

• (a) Using the procedure based on the Spearman sample correlation coefficient, find the P-value for a test of the hypothesis that starting position and finish position are independent variables against an appropriate alternative.

• (b) Repeat part (a) using the Kendall sample correlation coefficient. Are your findings similar?

• (c) Compare your results for this race with those obtained for the 1987 race in Exercise 11.B.7.

11.B.9. How Fast Is the Arctic Sea Ice Melting ? While the cause of climate change is bantered about in the popular media, the fact that it is occurring is not in question. Table 11.24 contains the extent of Arctic Sea Ice (in millions of square kilometers) in September for the years 1979 through 2012.

Table 11.24 Extent of Arctic Sea ice (millions of square kilometers) in September for the years 1979–2012

• (a) Plot the Arctic Sea Ice data versus the years of measurement.

• (b) Compute the Kendall sample correlation coefficient for the data in Table 11.24.

• (c) Find the P-value for testing the conjecture that there is a decreasing trend in the extent of Arctic Sea Ice in September over the period of time from 1979 through 2012.

• (d) Obtain the fitted least squares line for the data in Table 11.24. Plot this line on the plot of Arctic Sea Ice versus year of measurement. Does it look like a good fit?

• (e) Use this fitted least squares line to find the P-value for a hypothesis test of the conjecture that there is a linear decline in September Arctic Sea Ice over the period of time from 1979 through 2012.

11.B.10. Carbon Dioxide and Global Warming . One aspect of climate change that has received a lot of attention from the scientific community is the effect of atmospheric CO₂ (carbon dioxide) concentration on global temperature. Table 11.25 contains the atmospheric CO₂ concentration in parts per million and the Global Land-Ocean Temperature Index from the Goddard Institute of Space Studies (GISTEMP) for the years 1979–2010. GISTEMP is reported in units of 1/100 of a degree Centigrade increase above the 1950–1980 mean and is known in the literature as the global surface temperature anomaly.

Table 11.25 Atmospheric CO₂ concentration (parts per million) and GISTEMP (1/100 °C), 1979–2010

• (a) Compute both the Pearson correlation coefficient and the Spearman rank correlation coefficient between the atmospheric CO₂ concentration and GISTEMP.

• (b) Using the Pearson correlation coefficient, find the P-value for a hypothesis test of the conjecture that atmospheric CO₂ concentration and GISTEMP are positively correlated. Do the same for a hypothesis test using the Spearman rank correlation coefficient.

• (c) Obtain the fitted least squares line for the CO₂ concentration and GISTEMP data in Table 11.25.

• (d) Using the Kendall correlation coefficient, find the P-value for a hypothesis test of the conjecture that atmospheric CO₂ concentration was increasing over the period of time from 1979 to 2010.

11.1.3 11.C. Activities

11.C.1. Sodium and Calories in Canned Food . Go to your favorite supermarket and randomly select ten different canned food items from the shelves. For each of these items, record the amount of grams of sodium per serving and total calories per serving. Using these data, find the P-value for an appropriate test of the conjecture that grams of sodium and calorie content are positively correlated for canned foods.

11.C.2. Heart Rate and Blood Pressure . Obtain heart rate (in beats per minute) and systolic blood pressure (in mm. Hg) values for eight women and eight men.

• (a) Using all of the combined data for women and men, find the P-value for an appropriate test of the conjecture that heart rate and systolic blood pressure are positively correlated. Then carry out the same analysis separately for women and men. Discuss your findings.

• (b) Find the least squares fitted lines for the combined data and then separately for women and men. Discuss your findings.

• (c) Using the combined data, find the P-value for a test of the conjecture that there is a positive linear relationship between heart rate and systolic blood pressure.

11.C.3. Coffee and Bedtime . Survey at least 15 of your friends and/or classmates to obtain the following information from each of them: (i) average number of cups of coffee they drink in a 24 h day and (ii) their average bedtime, in minutes past ten p.m.

• (a) Using these data, find the P-value for the conjecture that these two variables are positively correlated.

• (b) Find the least squares fitted line for a linear regression of average bedtime on average daily cups of coffee. Find the P-value for an appropriate test of the significance of the linear regression.

11.1.4 11.D. Internet Archives

11.D.1. Grip Strength and Fraility . Chronological age is a natural marker of frailty. However, it is not a perfect marker, as there is wide variability in frailty between individuals of the same age. Numerous scientific studies have been conducted to investigate the possible connection between grip strength as a more reliable marker of frailty in older individuals. Use the Internet to find a scientific paper that addresses this association between grip strength and frailty. Summarize the findings discussed in the paper, particularly how the authors used correlation and regression to support their conclusions.

11.D.2. Passing Yardage and College Football Victories . Winning a college football game is dependent on a lot of performance variables. One of these variables is passing yardage. Use the Internet to find the following information for each of the Division I football games played on the most recent first Saturday in November:

• (i) Passing Yardage for the Winning Team

• (ii) Total Points Scored by the Winning Team

• (iii) Total Points Scored by the Losing Team.

• (a) Find the P-value for a test of the conjecture that total points scored by the winning team is positively correlated with the passing yardage for the winning team.

• (b) Find the P-value for a test of the conjecture that total points scored by the losing team is negatively correlated with the passing yardage of the winning team.

• (c) Find the least squares fitted line for the regression of total points scored by the winning team on the passing yardage for the winning team. Obtain the P-value for a test of the conjecture that there is a positive linear relationship between total points scored by the winning team and their passing yardage.

11.D.3. Rushing Yardage and College Football Victories . Use the Internet to find the rushing yardage for the winning team in each of the Division I football games played on the most recent first Saturday in November. Complete the following statistical analyses.

• (a) Find the P-value for a test of the conjecture that total points scored by the winning team is positively correlated with the rushing yardage for the winning team.

• (b) Find the P-value for a test of the conjecture that total points scored by the losing team is negatively correlated with the rushing yardage of the winning team.

• (c) Find the least squares fitted line for the regression of total points scored by the winning team on the rushing yardage for the winning team. Obtain the P-value for a test of the conjecture that there is a positive linear relationship between total points scored by the winning team and their rushing yardage.

• (d) Compare the results obtained in this exercise with those obtained in Exercise 11.D.2.

11.D.4. How Important are Three-Point Shots in College Basketball ? One of the more recent changes to the rules of college basketball has been the addition of the three-point arc, beyond which a made field goal counts for three points rather than the standard two points. While this is clearly an exciting option for the fans attending games, how much effect has it actually had on winning or losing basketball games? Use the Internet to find the following information for each of the Division I basketball games played on the most recent third Saturday in January:

• (i) Number of Made Three-Point Shots for the Winning Team

• (ii) Total Points Scored by the Winning Team.

• (a) Find the P-value for a test of the conjecture that total points scored by the winning team is positively correlated with the number of three-point shots they make.

• (b) Find the least squares fitted line for the regression of total points scored by the winning team on the number of three-point shots they make. Obtain the P-value for a test of the conjecture that there is a positive linear relationship between total points scored by the winning team and the number of three-point shots they make.

11.D.5. Trends in Never-Married Americans . The share of never-married Americans has been on the rise for the past five decades and men are more likely than women to have never been married. Use the Internet to find one or more reports that provide data to support this statement. Using these data, find the P-value for a test of the conjecture that there is a positive trend in the percentage of never-married American women over the period of time from 1960 through 2012. Do the same for never-married American men.

11.D.6. Explosion of Social Networking Sites . As of October 2015, nearly two-thirds of American adults were using at least one social networking site. How fast has been the rise in this acceptance of social networking? Use the Internet to find one or more reports that provide data to address this question. Using these data, find the P-value for a test of the conjecture that the use of social networking sites has been increasing since it stood at 7% in 2005. Are there differences in the rate of this rise due to age, gender, education, and income? Discuss the relevant findings from your report(s).

11.D.7. Shortage of Marriageable Men ? A Pew Research Center report found that over three-quarters of women surveyed cited that having a partner with a stable job was a very important attribute that they look for in someone to marry. But is that criterion becoming more difficult to satisfy? Use the Internet to find one or more reports that address this question over time for college-educated women who are 25–35 years old.