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Statistics Learning From Data

Thomas H. Short, Roxy Peck

Chapter 4

Describing Bivariate Numerical Data - all with Video Answers

Educators

Section 1

Correlation

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Problem 1

For each of the scatterplots shown, answer the following questions:
i. Does there appear to be a relationship between $x$ and $y ?$
ii. If so, does the relationship appear to be linear?
iii. If so, would you describe the linear relationship as positive or negative?

Emily Himsel
Emily Himsel
Numerade Educator
01:35

Problem 2

For each of the following pairs of variables, indicate whether you would expect a positive correlation, a negative correlation, or a correlation close to $0 .$ Explain your choice.
a. Interest rate and number of loan applications
b. Height and $\mathrm{IQ}$
c. Height and shoe size
d. Minimum daily temperature and cooling cost

Trent Speier
Trent Speier
Numerade Educator
02:40

Problem 3

The paper "The Relationship Between Cell Phone Use, Academic Performance, Anxiety, and Satisfaction with Life in College Students" (Computers in Human Behavior [2014]:
$343-350$ ) described a study of cell phone use among undergraduate college students at a large, Midwestern public university. The paper reported that the value of the correlation coefficient between $x=$ Cell phone use (measured as total amount of time (in hours) spent using a cell phone on a typical day) and $y=$ GPA (cumulative grade point average (GPA) determined from university records) was $r=-0.203$
a. Interpret the given value of the correlation coefficient. Does the value of the correlation coefficient suggest that students who use a cell phone for more hours per day tend to have higher GPAs or lower GPAs?
b. The study also investigated the correlation between texting (measured as the total number of texts sent and texts received per day) and GPA. The direction of the relationship between texting and GPA was the same as the direction of the relationship between cell phone use and GPA, but the relationship between texting and GPA was not as strong. Which of the following possible values for the correlation coefficient between texting and GPA could have been the one observed? $r=-0.30 \quad r=-0.10 \quad r=0.10 \quad r=0.30$
c. The paper included the following statement: "Participants filled in two blanks- one for texts sent and one for texts received. These two texting items were nearly perfectly correlated." Do you think that the value of the correlation coefficient for texts sent and texts received was close to $-1,$ close to $0,$ or close to + 1 ? Explain your reasoning.

Ivan Kochetkov
Ivan Kochetkov
Numerade Educator
08:33

Problem 4

The article "\$115K! The 13 Best Paying U.S. Companies" (USA TODAY, August 11,2015 ) gave the following data on median worker pay (in thousands of dollars) and the 1 -year percent change in stock price for the 13 highest paying companies in the United States.
a. Construct a scatterplot for these data.
b. Calculate and interpret the value of the correlation coefficient.
c. The article states that companies that pay more are seeing a payoff in their stock performance. Is this conclusion justified based on these data? Explain.
d. Is it reasonable to generalize conclusions based on these data to the population of all companies in the United States? Explain why or why not.

Jerelyn Nevil
Jerelyn Nevil
Numerade Educator
03:32

Problem 5

Each year, marketing firm J.D. Power and Associates surveys new car owners 90 days after they purchase their cars. These data are used to rate auto brands (Toyota, Ford, etc.) on quality and customer satisfaction. Data for 2015 on a quality rating (number of defects per 100 vehicles) and a satisfaction score for 30 brands sold in the United States are given in the accompanying table.
a. Construct a scatterplot of $y=$ Satisfaction rating versus $x=$ Quality rating. How would you describe the relationship between $x$ and $y ?$
b. Calculate and interpret the value of the correlation coefficient.

Mihir Nayar
Mihir Nayar
Numerade Educator
02:20

Problem 6

Is the following statement correct? Explain why or why not. A correlation coefficient of 0 implies that there is no relationship between two variables.

Blank Blank
Blank Blank
Numerade Educator
02:58

Problem 7

The article "That's Rich: More You Drink, More You Earn" (Calgary Herald, April 16,2002 ) reported that there was a positive correlation between alcohol consumption and income. Is it reasonable to conclude that increasing alcohol consumption will increase income? Explain why or why not.

Mathew Botros
Mathew Botros
Montclair State University
02:42

Problem 8

The paper "Noncognitive Predictors of Student Athletes' Academic Performance" (Journal of College Reading and Learning [2000]: e167) summarizes a study of 200 Division I athletes. It was reported that the correlation coefficient for college grade point average (GPA) and a measure of academic self-worth was $r=0.48$. Also reported were the correlation coefficient for college GPA and high school GPA $(r=0.46)$ and the correlation coefficient for college GPA and a measure of tendency to procrastinate $(r=-0.36) .$ Write a few sentences summarizing what these correlation coefficients tell you about GPA for the 200 athletes in the sample.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
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Problem 9

For each of the scatterplots shown, answer the following questions:
i. Does there appear to be a relationship between $x$ and $y ?$
ii. If so, does the relationship appear to be linear?
iii. If so, would you describe the linear relationship as positive or negative?

Emily Himsel
Emily Himsel
Numerade Educator
01:35

Problem 10

For each of the following pairs of variables, indicate whether you would expect a positive correlation, a negative correlation, or a correlation close to $0 .$ Explain your choice.
a. Weight of a car and gas mileage
b. Size and selling price of a house
c. Height and weight
d. Height and number of siblings

Trent Speier
Trent Speier
Numerade Educator
03:15

Problem 11

The paper "Digit Ratio as an Indicator of Numeracy Relative to Literacy in 7-Year-Old British Schoolchildren" (British Journal of Psychology [2008]: $75-85$ ) investigated a possible relationship between $x=$ Digit ratio (the ratio of the length of the second finger to the length of the fourth finger) and $y=$ Difference between numeracy score and literacy score on a national assessment. (The digit ratio is thought to be inversely related to the level of prenatal testosterone exposure.) The authors concluded that children with smaller digit ratios tended to have larger differences in test scores, meaning that they tended to have a higher numeracy score than literacy score. This conclusion was based on a correlation coefficient of $r=-0.22$. Does the value of the correlation coefficient indicate that there is a strong linear relationship? Explain why or why not.

James Kiss
James Kiss
Numerade Educator
00:55

Problem 12

Draw two scatterplots, one for which $r=1$ and a second for which $r=-1$.

Cory Kuzinski
Cory Kuzinski
Numerade Educator
04:17

Problem 13

Data from the U.S. Federal Reserve Board (www .federalreserve.gov/releases/housedebt/, retrieved April 21,
2017) on consumer debt (as a percentage of personal income) and mortgage debt (also as a percentage of personal income) for the 10 years from 2006 to 2015 are shown in the following table:
a. What is the value of the correlation coefficient for this data set?
b. Is it reasonable to conclude that there is no strong relationship between the variables (linear or otherwise)? Use a graphical display to support your answer.

Ria Yambao
Ria Yambao
Numerade Educator
01:38

Problem 14

The paper "Religiosity and Teen Birth Rate in the United States" (Reproductive Health [2009]: 14-20) included data on teen birth rate and on a measure of conservative religious beliefs for each of 49 U.S. states. Birth rate was measured as births per 1000 teenage females. The authors of the paper created a "religiosity" score for each state by using responses to a large national survey conducted by the Pew Forum on Religion and Public Life. Higher religiosity scores correspond to more conservative religious beliefs. The authors reported that "teen birth rate is very highly correlated with religiosity at the state level, with more religious states having a higher rate of teen birth." The following data for 49 states are from a graph in the paper.
a. Construct a scatterplot of $y=$ Birth rate versus $x=$ Religiosity score. How would you describe the relationship between $x$ and $y$ ? What aspects of the scatterplot support what the authors concluded about teen birth rate and religiosity?
b. Calculate and interpret the value of the correlation coefficient.

Qudsiya Anis
Qudsiya Anis
Numerade Educator
01:01

Problem 15

Each year the Harris Poll surveys Americans on a number of issues. It uses responses to several questions to calculate a "Happiness" index that measures overall happiness. The article "Latest Happiness Index Reveals American Happiness at All-Time Low" (www.theharrispoll.com/health -and-life/American-Happiness_at-All-Time-Low.html, retrieved April 21,2017 ) included the happiness index for the seven years between 2008 and 2016 . Also included in the article were the percentages of people who responded "Somewhat Agree" or "Strongly Agree" to the following statements: Statement 1 (Happy with Life) 1: At this time, I'm generally happy with my life. Statement 2 (Won't Benefit): I won't get much benefit from the things that I do anytime soon.
a. Calculate the value of the correlation coefficient for Happiness index and the response to the Happy with Life statement.
b. Calculate the value of the correlation coefficient for Happiness index and the response to the Won't Benefit statement.
c. Is there a stronger relationship between Happiness index and the response to Statement 1 or between Happiness index and the response to Statement $2 ?$
d. Write a few sentences describing the relationships between Happiness index and the responses to the two statements.

Nick Johnson
Nick Johnson
Numerade Educator
01:22

Problem 16

The amount of money spent each year on science, space, and technology in the United States (in millions of dollars) and the amount of money spent on pets in the United States (in billions of dollars) for the years 2000 to 2009 were used to construct the following graph. (The data are from the web site www.tylervigen.com/spurious-correlations, retrieved August $28,2016 .)$

Rupsa Sarkar
Rupsa Sarkar
Numerade Educator
01:28

Problem 17

Below are the data used to construct the time series plots in the previous exercise. Calculate the value of the correlation coefficient for the amount spent on science, space, and technology and the amount spent on pets. Explain how this value is consistent with your answer to the previous exercise.

Tawana Stiff
Tawana Stiff
Numerade Educator
01:34

Problem 18

An auction house released a list of 25 recently sold paintings. The artist's name and the sale price of each painting appear on the list. Would the correlation coefficient be an appropriate way to summarize the relationship between artist and sale price? Why or why not?

Kaylee Mcclellan
Kaylee Mcclellan
Numerade Educator
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Problem 19

A sample of automobiles traveling on a particular segment of a highway is selected. Each one travels at roughly a constant rate of speed, although speed does vary from auto to auto. Let $x=$ Speed and $y=$ Time needed to travel this segment. Would the sample correlation coefficient be closest to $0.9,0.3,-0.3,$ or $-0.9 ?$ Explain.

Jason Gerber
Jason Gerber
Numerade Educator
05:01

Problem 20

It may seem odd, but biologists can tell how old a lobster is by measuring the concentration of pigment in the lobster's eye. The authors of the paper "Neurolipofuscin Is a Measure of Age in Panulirus argus, the Caribbean Spiny Lobster, in Florida" (Biological Bulletin [2007]: 55-66) wondered if it was sufficient to measure the pigment in just one eye, which would be the case if there is a strong relationship between the concentration in the right eye and the concentration in the left eye. Pigment concentration (as a percentage of tissue sample) was measured in both eyes for 39 lobsters, resulting in the following summary quantities (based on data from a graph in the paper):
$$\begin{array}{ccc}n=39 & \Sigma x=88.8 & \Sigma y=86.1 \\ \Sigma x y=281.1 & \Sigma x^{2}=288.0 & \Sigma y^{2}=286.6\end{array}$$
An alternative formula for calculating the correlation coefficient that doesn't involve calculating the z-scores is $$r=\frac{\Sigma x y-\frac{(\Sigma x)(\Sigma y)}{n}}{\sqrt{\Sigma x^{2}-\frac{(\Sigma x)^{2}}{n}} \sqrt{\Sigma y^{2}-\frac{(\Sigma y)^{2}}{n}}}$$
Use this formula to calculate the value of the correlation coefficient, and interpret this value.

Lucas Finney
Lucas Finney
Numerade Educator