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Elementary Statistics

Robert Johnson, Patricia Kuby

Chapter 13

Linear Correlation and Regression Analysis - all with Video Answers

Educators

Section 1

Linear Correlation Analysis

03:33

Problem 1

Consider the "Married Couples' Heights" scatter diagram presented in "Height Compatibility" on page 612 .
a. What is the independent variable for this set of data? How does it appear on the scatter diagram?
b. What is the dependent variable for this set of data? How does it appear on the scatter diagram?
c. Does there appear to be a relationship between wife's and husband's heights? How does this appear on the scatter diagram?

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
09:08

Problem 2

Consider the "Married Couples' Heights" scatter diagram presented in "Height Compatibility" on page 612:
a. How can a husband and wife of the same height be located on the given scatter diagram? How many were there? (Determine this answer in two ways: $1-$ count the points on the scatter diagram; $2-$ count the ordered pairs in the data set. Explain the discrepancy.)
b. How can husbands who are taller than their wives be located on the scatter diagram? How many were there?
c. How can husbands who are shorter than their wives be located on the scatter diagram? How many were there?
d. Have you accounted for all 87 couples listed in the data? Verify.
e. How does the greatest difference in heights appear on the scatter diagram?

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
07:18

Problem 3

Consider a set of paired bivariate data.
a. $\quad$ Explain why $\Sigma(x-\bar{x})=0$ and $\Sigma(y-\bar{y})=0$.
b. $\quad$ Describe the effect that lines $x=\bar{x}$ and $y=\bar{y}$ have on the graph of these points.
c. $\quad$ Describe the relationship of the ordered pairs that will cause $\Sigma[(x-\bar{x}) \cdot(y-\bar{y})]$ to be (1) positive, (2) negative, and (3) near zero.

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
01:15

Problem 4

Matches correlation coefficients with their scatterplots. After several practice rounds using "New Plots," explain your method of matching.

Arwa Ali
Arwa Ali
Numerade Educator
05:39

Problem 5

The following data values are from a random sample of 40 college students; the data show the students' gender, American College Test (ACT) composite scores, and grade point averages (GPAs) after their first term in college.
a. Construct a scatter diagram of the data with ACT scores on the horizontal axis and GPA on the vertical axis, being sure to identify the male and female students.
b. Do the patterns for males and females appear to be the same, or are they different? Identify specific similarities and differences.
c. Assume that a student had an ACT score of $25 .$ What would you predict that student's GPA to be at the end of the first term in college?
d. Does there appear to be any relationship between ACT scores and the first-term GPA?

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
01:37

Problem 6

Aerial photographs are one of many techniques used to monitor wildlife populations. Knowing the number of animals and their locations relative to areas inhabited by the human population is very useful. It is also important to monitor physical characteristics of the animals. Is it possible to use the length of a bear, as estimated from an aerial photograph, to estimate the bear's age and/or weight? (It would be much safer than asking it to stand on a set of scales! @) The data that follow are for age (in months), gender $(1=$ male, $2=\text { female }),$ length (in inches), and weight (in pounds).
a. Investigate the relationship between the length and age of the bears. Be sure to include the gender variable.
b. Does there seem to be a predictable pattern for the relationship between length and age? How does gender of the bear affect the relationship? Explain. Describe the pattern.
c. Investigate the relationship between the length and weight of the bears. Be sure to include the gender variable.
d. Does there seem to be a predictable pattern for the relationship between length and weight? How does gender of the bear affect the relationship? Explain. Describe the pattern.
e. If the gender of a smaller or younger bear cannot be determined, how will this affect the estimate for age or weight? Explain.

Nick Johnson
Nick Johnson
Numerade Educator
18:13

Problem 7

a. Construct a scatter diagram of the following bivariate data.
$$\begin{array}{lcccccccccc}
\text { Point } & \mathbf{A} & \mathbf{B} & \mathbf{C} & \mathbf{D} & \mathbf{E} & \mathbf{F} & \mathbf{G} & \mathbf{H} & \mathbf{I} & \mathbf{J} \\
\hline \mathbf{x} & 1 & 1 & 3 & 3 & 5 & 5 & 7 & 7 & 9 & 9 \\
\mathbf{y} & 1 & 2 & 2 & 3 & 3 & 4 & 4 & 5 & 5 & 6 \\
\hline
\end{array}$$
b. Calculate the covariance.
c. Calculate $s_{x}$ and $s_{y}$.
d. Calculate $r$ using formula (13.2).
e. Calculate $r$ using formula (13.3).

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
13:03

Problem 8

a. Draw a scatter diagram of the following bivariate data.
$$\begin{array}{lllllllllll}
\text { Point } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { F } & \text { G } & \text { H } & \text { I } & \text { J } \\
\hline \mathbf{x} & 0 & 1 & 1 & 2 & 3 & 4 & 5 & 6 & 6 & 7 \\
\mathbf{y} & 6 & 6 & 7 & 4 & 5 & 2 & 3 & 0 & 1 & 1 \\
\hline
\end{array}$$
b. Calculate the covariance.
c. Calculate $s_{x}$ and $s_{y}$.
d. Calculate $r$ using formula ( 13.2 ).
e. Calculate $r$ using formula (13.3).

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
07:00

Problem 9

A computer was used to complete the preliminary calculations; form the extensions table; calculate the summations $\Sigma x, \Sigma y, \Sigma x^{2}, \Sigma x y, \Sigma y^{2}$; and find the $\operatorname{SS}(x), \operatorname{SS}(y),$ and $\operatorname{SS}(x y)$ for the following set of bivariate data. Verify the results by calculating the values yourself.
$$\begin{array}{lllllll}
\hline x & 45 & 52 & 49 & 60 & 67 & 61 \\
y & 22 & 26 & 21 & 28 & 33 & 32 \\
\hline
\end{array}$$

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
03:39

Problem 10

Use a computer to form the extensions table; calculate the summations $\Sigma x, \Sigma y, \Sigma x^{2}, \Sigma x y, \Sigma y^{2}$; and find the $\operatorname{SS}(x), \operatorname{SS}(y),$ and $\operatorname{SS}(x y)$ for the following set of bivariate data.
$$\begin{array}{lrrrrrrrr}
\hline x & 11.4 & 9.4 & 6.5 & 7.3 & 7.9 & 9.0 & 9.3 & 10.6 \\
y & 8.1 & 8.2 & 5.8 & 0.4 & 5.9 & 0.5 & 7.1 & 7.8 \\
\hline
\end{array}$$

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
06:34

Problem 11

NFL football enthusiasts often look at a team's total points scored for (Pts $\mathrm{F}$ ) and total points scored against (Pts $A$ ) as a way of comparing the relative strength of teams. The season totals for the 32 teams in the NFL's 2009 season are shown below:
$$\begin{array}{cc|cc|cc|cc}
\text { Pts } \mathbf{F} & \text { Pts } \mathbf{A} & \text { Pts } \mathbf{F} & \text { Pts } \mathbf{A} & \text { Pts } \mathbf{F} & \text { Pts } \mathbf{A} & \text { Pts } \mathbf{F} & \text { Pts } \mathbf{A} \\
\hline 427 & 285 & 305 & 291 & 416 & 307 & 454 & 320 \\
348 & 236 & 391 & 261 & 388 & 333 & 326 & 324 \\
360 & 390 & 368 & 324 & 354 & 402 & 197 & 379 \\
258 & 326 & 245 & 375 & 290 & 380 & 294 & 424 \\
361 & 250 & 470 & 312 & 510 & 341 & 375 & 325 \\
429 & 337 & 461 & 297 & 363 & 325 & 330 & 281 \\
402 & 427 & 327 & 375 & 315 & 308 & 280 & 390 \\
266 & 336 & 262 & 494 & 244 & 400 & 175 & 436 \\
\hline
\end{array}$$
a. Calculate the linear correlation coefficient (Pearson's product moment, $r$ ) for the points scored for and against.
b. What conclusion can you draw from the answer in part a?
c. Construct the scatter diagram and comment on how it supports, or disagrees with, your comments in part b.

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
07:56

Problem 12

Knowing a horse's weight (measured in pounds) is important information for a horse owner. The amount of feed and medicine dosages all depend on the horse's weight. Most owners do not have the resources to have a scale large enough to weigh a horse, so other measurements are used to estimate the weight. Height (measured in hands) and girth and length (measured in inches) are common measurements for a horse. A sample of Suffolk Punch stallion measurements were taken from the website http://www.suffolkpunch.com/.
$$\begin{array}{ccccc}
\text { Row } & \text { Height } & \text { Girth } & \text { Length } & \text { Weight } \\
\hline 1 & 16.0 & 93 & 72 & 1825 \\
2 & 15.3 & 78 & 69 & 1272 \\
3 & 16.0 & 84 & 70 & 1515 \\
4 & 17.0 & 90 & 80 & 2100 \\
5 & 16.2 & 86 & 70 & 1569 \\
6 & 16.0 & 88 & 72 & 1690 \\
7 & 16.0 & 83 & 72 & 1500 \\
\hline
\end{array}$$
a. Calculate the linear correlation coefficient (Pearson's product moment, $r$ ) between (1) height and weight, (2) girth and weight, and ( 3 ) length and weight.
b. What conclusions might you draw from your answers in part a?
c. Construct a scatter diagram for each pair of variables listed in part a.
d. Do the scatter diagrams support your answer in part b?
e. Based on this evidence, which measurement do you believe has the most potential as a predictor of weight? Explain your choice.

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
07:33

Problem 13

a. Calculate the covariance of the set of data $(20,10),(30,50),(60,30),(80,20),(110,60),$ and (120,10).
b. Calculate the standard deviation of the six $x$ values and the standard deviation of the six $y$ values.
c. Calculate $r,$ the coefficient of linear correlation, for the data in part a.
d. Compare these results to those found in the text for Example 13.1 (pp. $613-616$ ).

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
08:38

Problem 14

A formula that is sometimes given for computing the correlation coefficient is
$$r=\frac{n\left(\sum x y\right)-\left(\sum x\right)\left(\sum y\right)}{\sqrt{n\left(\sum x^{2}\right)-\left(\sum x\right)^{2}} \sqrt{n\left(\sum y^{2}\right)-\left(\sum y\right)^{2}}}$$
Use this expression as well as the formula
$$r=\frac{\mathrm{SS}(x y)}{\sqrt{\mathrm{SS}(x) \cdot \mathrm{SS}(y)}}$$
to compute $r$ for the data in the following table.
$$\begin{array}{llllll}
x & 2 & 4 & 3 & 4 & 0 \\
\hline y & 6 & 7 & 5 & 6 & 3
\end{array}$$

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator