Deck 14: Descriptive Methods in Regression and Correlation

Question 1

Provide an appropriate response.
A regression equation is obtained for a set of data. After examining a scatterplot, the
researcher notices a data point that is potentially an influential observation. How could the
researcher confirm that this data point is indeed an influential observation? How should
the researcher proceed if the data point is found to be an influential observation?

Answer 1

To confirm that the data point is an influential observation, the researcher should
remove that data point and recalculate the regression equation. If the new regression
equation differs markedly from the original regression equation, the researcher can
conclude that the point is an influential observation. The researcher should then try
to determine the reason for the influential observation. If it is due to a measurement
or recording error, the researcher should remove the data point from the data set. If
it is a legitimate data point; the researcher may need to remove it nonetheless or
obtain additional data, so that the regression analysis is not so dependent on one
data point. (Explanations will vary.)

Question 2

Solve the problem.
A ball is thrown downward from a tall building with an initial velocity of 10 meters per
second (m/sec). According to the laws of physics, if you let y denote the velocity of the ball
after x seconds, y = 10 + 9.8x. Determine b0 and b1 for this linear equation. Determine the
velocity of the ball after 1, 2, 3, and 4 seconds. Use these four points to graph the linear
equation y = 10 + 9.8x. Use the graph to estimate visually the velocity of the ball after 2.5
seconds.

Answer 2

Velocity of the ball after 2.5 seconds is 34.5 m/sec. Estimates should be close to this.

Question 3

Provide an appropriate response.
A regression equation is obtained for the following set of data. For what range of x-values would it be reasonable to use the regression equation to predict
the y-value corresponding to a given x-value? Why?

Answer 3

It would be reasonable to use the regression equation to predict the y-value
corresponding to a given x-value for x-values in the range from 2 to 12. We can
reasonably use the regression equation to make predictions for values of the
predictor variable (x)within the range of the observed values of the predictor value,
in this case from 2 to 12. However, to do so for values of the predictor variable
outside that range may not be reasonable, because the linear relationship between
the variables may not hold there. Using the regression equation to make predictions
for values of the predictor variable outside the range of the observed values of the
predictor variable is called extrapolation. Grossly incorrect predictions can result
from extrapolation. (Explanations will vary.)

Question 4

Solve the problem.
For a compact car, a car-rental company charges $28.50 per day plus $0.15 per mile. For a
one-day rental, let x denote the number of miles driven and let y denote the total cost.
Obtain the equation that expresses y in terms of x. Construct a table of values using the
x-values 70, 140, and 220 miles. Draw the graph of the equation by plotting the points from
the table and connecting them with a straight line. Use the graph to estimate visually the
cost of driving the car 190 miles.

Question 5

Solve the problem.
A car mechanic tells a client that it will cost $120 for parts plus $50 per hour for labor to fix
her car. Let x denote the number of hours of labor and let y denote the total cost to fix the
car. Obtain the equation that expresses y in terms of x. Construct a table of values using the
x-values 2, 3, and 5 hours. Draw the graph of the equation by plotting the points from the
table and connecting them with a straight line. Use the graph to estimate visually the total
cost of fixing the car if the number of hours of labor is 3.5.

Question 6

Solve the problem.
Anne is running a 400-meter race. She runs at a constant speed of 7.5 meters per second. If
you let y denote her distance in meters from the finish line x seconds after the start of the
race, y = 400 - 7.5x. Determine b0 and b1 for this linear equation. Find Anne's distance
from the finish line 10, 24, and 43 seconds after the race begins. Use these three points to
graph the linear equation y = 400 - 7.5x. Use the graph to estimate visually Anne's distance
from the finish line 32 seconds after the start of the race.

Question 7

Provide an appropriate response.
Suppose data are collected for each of several randomly selected adults for height, in
inches, and number of calories burned in 30 minutes of walking on a treadmill at 3.5 mph.
How would the value of the linear correlation coefficient, r, change if all of the heights
were converted to meters?

Question 8

Provide an appropriate response.
What is the relationship between the linear correlation coefficient and the usefulness of the
regression equation for making predictions?

Question 9

Provide an appropriate response.
For each of 200 randomly selected cities, Pete compared data for the number of churches in
the city (x)and the number of homicides in the past decade (y). He calculated the linear
correlation coefficient and was surprised to find a strong positive linear correlation for the
two variables. Does this suggest that when a city builds new churches this will tend to
cause an increase in the number of homicides? Why do you think that a strong positive
linear correlation coefficient was obtained?

Question 10

Provide an appropriate response.
Create a scatterplot that shows a perfect positive linear correlation between x and y. How
would the scatterplot change if the correlation showed each of the following?
(a)a strong positive linear correlation;
(b)a weak positive linear correlation;
(c)no linear correlation.

Question 11

Provide an appropriate response.
Explain why having a high linear correlation does not imply causality. Give an example to
support your answer.

Question 12

Provide an appropriate response.
Give an example of a linear equation whose graph is a horizontal line.

Question 13

Provide an appropriate response.
Define the terms "predictor variable" and "response variable." Give an example of each.

Question 14

Provide an appropriate response.
The variables height and weight could reasonably be expected to have a positive linear
correlation coefficient, since taller people tend to be heavier, on average, than shorter
people. Give an example of a pair of variables which you would expect to have a negative
linear correlation coefficient and explain why. Then give an example of a pair of variables
whose linear correlation coefficient is likely to be close to zero.

Question 15

Provide an appropriate response.
Determine which scatterplot shows the strongest linear correlation.

Question 16

Solve the problem.
For a day's work, Chris is paid $50 to cover expenses plus $16 per hour. Let x denote the
number of hours Chris works in a day and let y denote Chris's total salary for the day.
Obtain the equation that expresses y in terms of x. Construct a table of values using the
x-values 2, 4, and 8 hours. Draw the graph of the equation by plotting the points from the
table and connecting them with a straight line. Use the graph to estimate visually Chris's
salary for the day if he works 6 hours.

Question 17

Provide an appropriate response.

Question 18

Provide an appropriate response.
For a particular regression analysis, it is found that SST = 901 and SSE = 804.1. Does the
regression equation appear to be useful for making predictions? How can you tell?

Question 19

Provide an appropriate response.
For a certain linear equation, as x increases from 3 to 4, the y-value increases from 17 to 22.
The y-value corresponding to an x-value of 9 is 47. What is the y-value corresponding to
an x-value of 10? Explain how you solved this problem.

Question 20

Provide an appropriate response.
Describe what scatterplots are, and discuss their importance.

Question 21

Question 22

Obtain the linear correlation coefficient for the data. Round your answer to three decimal places.

Question 23

You are given information about a straight line. Use two points to graph the equation.

Question 24

Provide an appropriate response.

Question 25

The y-intercept and slope, respectively, of a straight line are given. Find the equation of the line.
-9 and 4

Question 26

Compute the specified sum of squares.

Question 27

Compute the coefficient of determination. Round your answer to four decimal places.
For a particular regression analysis, it is found that SST = 913.5 and SSE = 277.9.

A)0.3042
B)0.8341
C)3.2872
D)0.6958

Question 28

Provide an appropriate response.
True or false? In the context of regression analysis, if the regression sum of squares is large relative
to the error sum of squares, then the regression equation is useful for making predictions.

Question 29

Provide an appropriate response.
When performing regression analysis, how can you evaluate how useful the regression
equation is for making predictions?

Question 30

Determine the y-intercept and slope of the linear equation.

Question 31

Provide an appropriate response.

Question 32

Provide an appropriate response.

Question 33

The y-intercept and slope, respectively, of a straight line are given. Find the equation of the line.

Question 34

The regression equation for the given data points is provided. Graph the regression equation and the data points.^y

Question 35

Obtain the linear correlation coefficient for the data. Round your answer to three decimal places.

Question 36

You are given information about a straight line. Determine whether the line slopes upward, slopes downward, or ishorizontal.
The y-intercept is 0 and the slope is -5.2.

A)Is horizontal
B)Slopes upward
C)Slopes downward

Question 37

Provide an appropriate response.

Question 38

Provide an appropriate response.

Question 39

Provide an appropriate response.
True or false? In the context of regression analysis, the regression sum of squares is the variation in
the observed values of the response variable explained by the regression.

Question 40

Is the data point, P, an outlier, a potential influential observation, both, or neither?

Question 41

Determine the percentage of variation in the observed values of the response variable that is explained by the regression.Round to the nearest tenth of a percent if needed.

Question 42

The y-intercept and slope, respectively, of a straight line are given. Find the equation of the line.

Question 43

Compute the coefficient of determination. Round your answer to four decimal places.

Question 44

Determine the y-intercept and slope of the linear equation.
y = 1

A)y-intercept = -1, slope = 0
B)y-intercept = 1, slope = 0
C)y-intercept = 0, slope = 1
D)y-intercept = 1, slope = 1

Question 45

Solve the problem.

Question 46

Provide an appropriate response.

Question 47

Obtain the linear correlation coefficient for the data. Round your answer to three decimal places.
Managers rate employees according to job performance (x)and attitude (y). The results for several randomly selected employees are given below.

Question 48

You are given information about a straight line. Determine whether the line slopes upward, slopes downward, or ishorizontal.

Question 49

Provide an appropriate response.
The table below shows the age and annual income of 12 randomly selected college graduates all living in the city of Seattle. Would it be reasonable to use the regression equation to predict the annual income of a college
Graduate in Seattle who is 90 years old? Explain your answer.

A)No; 90 year olds are outside the age range of the data.
B)No; the regression line does not fit the data very closely.
C)No; regression equations can not be used to predict values for which there is no input data.
D)Yes; the regression line fits the data quite closely.

Question 50

You are given information about a straight line. Use two points to graph the equation.
The equation of the line is y = 1 - 0.75x.

Question 51

You are given information about a straight line. Determine whether the line slopes upward, slopes downward, or ishorizontal.
The equation of the line is y = -5.6 - 6x.

A)Slopes downward
B)Is horizontal
C)Slopes upward

Question 52

Obtain the linear correlation coefficient for the data. Round your answer to three decimal places.
The data below show the cost of advertising (x), in thousands of dollars, and the number of products sold (y), in thousands, for each of eight randomly selected product lines.

Question 53

Determine the regression equation for the data. Round the final values to three significant digits, if necessary.

Question 54

Obtain the linear correlation coefficient for the data. Round your answer to three decimal places.
Two separate tests, x and y, are designed to measure a student's ability to solve problems. Several students are randomly selected to take both tests and their results are shown below.

Question 55

Provide an appropriate response.
True or false? In the context of regression analysis, the coefficient of determination is the proportion
of variation in the observed values of the response variable not explained by the regression