Deck 16: Regression

Question 1

The slope of the true regression line is given by

A) $\alpha$ .
B) $\beta$ .
C) $a$ .
D) $b$ .

Answer 1

$\beta$ .

Question 2

In regression analysis, the quantity that gives the amount by which $y$ changes for a unit change in $x$ is called the

A) $y$ -intercept.
B) slope.
C) standard error of estimates.
D) standard deviation.

Answer 2

slope.

Question 3

The symbol used to indicate the $y$ values of points on a least-squares line is

A) $b$ .
B) $\bar{y}$ .
C) $\hat{y}$ .
D) $y$ .

Answer 3

$\hat{y}$ .

Question 4

The normal equations are obtained from minimizing

A) $\sum(y-\hat{y})^{2}$ .
B) $\sum(y-\bar{y})^{2}$ .
C) $\sum(y-y)$ .
D) $\sum(\bar{y}-y)$ .

Question 5

The symbol $a$ in the least-squares equation is a sample estimate of

A) $\alpha$ .
B) $\beta$ .
C) $\mu_{x \mid y}$ .
D) $s_{e}$ .

Question 6

If we are testing the hypothesis that $\beta=0$ against the alternative that $\beta \neq 0$ when there are 12 data points and $\alpha=0.05$ , the correct $t$ value to be used is

A) 1.812 .
B) 1.796 .
C) 2.228 .
D) 2.201 .

Question 7

The symbol $b$ in the least-squares equation is a sample estimate of

A) $\alpha$ .
B) $\beta$ .
C) $\mu_{y} \mid x$ .
D) $s_{e}$ .

Question 8

Given the regression equation $y=3+7 x$ , the most central value in a confidence interval for the mean of $y$ , given $x=4$ is

A) 4 .
B) 31 .
C) 7 .
D) 3 .

Question 9

The symbol that represents the true regression line is

A) $y$ .
B) $\mu_{y} \mid x_{0}$ .
C) $\beta$ .
D) $\mu_{y} \mid x$ .

Question 10

The width of a confidence interval for $\mu_{y} \mid x_{o}$ will be increased if

A) $x_{0}$ is moved closer to $\bar{x}$ .
B) $s_{e}$ is decreased.
C) $n$ is increased.
D) $t$ is increased.

Question 11

The advertising expense and profit of a company in thousands of dollars for each of five years is given below:
a. Solve the normal equations to find the equation of the least-squares line which will allow us to predict profit from advertising expense.
b. Check your values of $a$ and $b$ using the "solutions of normal equation."
c. If advertising expense is to be $\$ 9,000$ for a particular year, predict the profit for that year.

Question 12

A personnel manager wants to predict the salary (in thousands of dollars) of a systems analyst based on number of years of experience. A random sample of 12 systems analysts produces the following results:

a. Find the least-squares line which predicts salary based on years of experience.
b. Predict the salary of a system analyst with five years experience.

Question 13

Table 16.1
The table below shows annual profit figures (in thousands of dollars) for a company.

-Fit an exponential trend to the data in Table 16.1.

Question 14

Table 16.1
The table below shows annual profit figures (in thousands of dollars) for a company.

-Fit a power curve to the data in Table 16.1.

Question 15

The following data gives the demand of a product (in hundreds of units) for five different price levels (in dollars):
Fit a parabola to this data.

Question 16

Table 16.2
The data gives the number of $18-25$ year olds (for the given years) whose families earn between $\$ 15-\$ 25,000$ per year $(X)$ and the number of U.S. college students for the given year ( $Y$ ).

-Use the data in Table 16.2 to:
a. Find the least-squares regression equation which predicts $Y$ from $X$ .
b. Find the standard error of estimate.
c. Predict $Y$ if $X=4.00$ .

Question 17

Table 16.3
A study is supposed to examine the relationship between the amount spent on advertising a new product $(x)$ and consumer awareness of the product ( $y$ ) based on the proportion of people who have heard of it. Suppose a sample shows the data below for four different products.

-Use the data in Table 16.3 to solve the following:
a. Find the equation of the least-squares line that will allow us to predict consumer awareness from the advertising expense.
b. Calculate the standard error of estimate.
c. Can the equation found in part (a) be used for an advertising expense of 60 thousand dollars? Why or why not?

Question 18

Using a 95% confidence level, construct an interval estimate of the mean consumer awareness of a single product, assuming $\$ 700,000$ is spent for its promotion.

Question 19

Using a $5 \%$ significance level, test the hypothesis that advertising expenditure has no impact on consumer awareness of a single product, assuming $\$ 700,000$ is spent for its promotion.

Question 20

The estimated regression line is the line that minimizes the sum of the squares of the distances from the given points to the line.

Question 21

The symbol $y$ gives the true mean of $y$ for a given value of $x=x_{0}$ .

Question 22

The standard deviation measures the dispersion of the $y$ values about the estimated least-squares line.

Question 23

A confidence interval for a future individual value is wider than a confidence interval for a mean of $y$ when $x=x_{0}$ .

Question 24

In regression analysis, the variable that we are trying to predict is called the independent variable.

Question 25

Multiple regression involves at least two dependent variables.

Question 26

In general, the smaller the standard error of estimate the better the least-squares regression line.

Question 27

The estimated and true regression lines are always the same.

Question 28

If the slope of the true regression line is zero, then the slope of the sample regression line may be different from zero.

Question 29

If paired data plotted on log-log paper fall close to a straight line, we would expect a parabola to provide a good fit for the data.

Question 30

When we make inferences about the regression coefficients $\alpha$ and $\beta$ , the number of degrees of freedom that is required to use the $t$ table is __________.

Question 31

The term which measures the dispersion of the $y$ values about the estimated least-squares line is called __________.

Question 32

In linear regression analysis, we assume that for each value of $x$ the variable to be predicted has a mean of __________.

Question 33

The symbols used for the estimated regression coefficients when there is one independent variable are __________ and __________

Question 34

In a test of the null hypothesis that $\beta=10$ against the alternative hypothesis that $\beta>10$ , the null hypothesis will be rejected if the obtained $t$ value is __________ the tabled $t$ value.

Question 35

When we use observed data to derive a mathematical equation and use it to predict the value of one variable from a given value of another, the procedure is known as __________.

Question 36

The symbol for the mean of $y$ when $x=x_{0}$ is __________.

Question 37

In a two-tailed hypothesis test that $\beta=0$ , the tabled $t$ values are -2.262 and 2.262 . If the obtained $t$ value is 1.83 , the null hypothesis __________ rejected.

Question 38

A multiple regression equation with three independent variables has the form __________.

Question 39

If paired data plotted on semi-log paper fall close to a straight line, then we would expect a (an) __________.