Deck 17: Correlation

Question 1

A statistic that measures the relationship between variables while eliminating the effects of other variables is called the

A) correlation coefficient.
B) regression coefficient.
C) standard error of estimate.
D) partial correlation coefficient.

Answer 1

partial correlation coefficient.

Question 2

The sum $\sum(\hat{y}-\bar{y})^{2}$ is called the __________ sum of squares.

A) regression
B) total
C) at least
D) residual

Answer 2

regression

Question 3

The sum $\sum(y-\bar{y})^{2}$ is called the __________ sum of squares.

A) regression
B) total
C) at least
D) residual

Answer 3

total

Question 4

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

A) partial correlation coefficient.
B) slope of the regression line.
C) $y$ -intercept of the regression line.
D) correlation coefficient.

Question 5

The correlation coefficient is equal to

A) the total variation.
B) the unexplained variation.
C) one minus the proportion of unexplained variation.
D) the square root of the explained variation.

Question 6

If the hypothesis that $Q=0$ is rejected, then

A) $s_{e}$ must be zero.
B) $r^{2}$ must be close to one.
C) the regression slope may be close to zero.
D) there is a strong relationship between $x$ and $y$ .

Question 7

Table 17.1
The mathematics S.A.T. scores and college grade point averages of eight students are given below.


-Use Table 17.1 to solve the following:
a. Calculate the correlation coefficient $r$ .
b. What is the proportion of total variation in $y$ that is accounted for by $x$ ?
c. Test the null hypothesis of no relationship between the two variables at $\alpha=0.05$ .
d. Calculate a $95 \%$ confidence interval for $\mathrm{Q}$ .

Question 8

In a multiple regression problem, the regression sum of squares is $\sum(\hat{y}-\bar{y})^{2}=12,000$ and the total sum of squares is $\sum(y-\bar{y})^{2}=20,000$ . Find the value of the multiple correlation coefficient.

Question 9

Table 17.2
Below is a list of countries with their 1979 "excess monetary growth*" ( along with their mid -1980 inflation rate).
*Excess monetary growth is the extent to which money supply growth exceeds to the growth of constant dollar GNP; specifically, the ratio of 1979 M2 to 1978 M2 is divided by the ratio of 1979 real GNP to 1978 real GNP.

-Use Table 17.2 to solve the following:
a. Calculate the correlation coefficient.
b. What is the proportion of the total variation in inflation rate that is explained by excess monetary growth?
c. Test the null hypothesis that $\mathrm{Q}=0.3$ at $\alpha=0.01$ .
d. Calculate a $99 \%$ confidence interval for $\mathrm{Q}$ .

Question 10

Table 17.3
The table below gives the number of housing starts and the unemployment rates for a sequence of quarters:


-Use Table 17.3 to solve the following:
a. Calculate the correlation coefficient.
b. What is the proportion of the total variation in unemployment rates that is explained by the number of housing starts?
c. Test the null hypothesis that $\mathrm{Q}=-0.1$ at $\alpha=0.05$ .

Question 11

A financial economist wants to evaluate how interest rates affect the inflation rate. Here are some results on yearly prime interest rates $(x)$ and inflation rates $(y)$ for the 13 years 1965 through 1977. $\sum x y=500, \sum y=78$ , $\sum x^{2}=450, \sum y^{2}=600, \sum x=65$ .
For the situation above:
a. Calculate the sample coefficient of determination and correlation coefficient.
b. Give the proportion of variation explained by the regression line.
c. Test the null hypothesis of no correlation against a two-tailed alternative at the 0.01 significance level.

Question 12

For two sets of data $x$ and $y, r=0.75$ . Suppose each value of $x$ is measured in feet. If each $x$ value is converted to inches by multiplying by 12, what is the new value of $r$ ? Suppose, instead, a constant is added to each value of $x$ . What is the new $r$ value?

Question 13

The variables circulation $\left(x_{1}\right)$ and advertising income $\left(x_{2}\right)$ are used to predict the annual profit of a daily newspaper ( $y$ ). A multiple correlation coefficient of 0.65 is obtained. For predicting annual profit on the basis of advertising income alone, a value of $r=0.78$ is found. Comment on these results.

Question 14

If $r$ has a value of 0.60 , then $60 \%$ of the variation in $y$ can be explained by the variation in $x$ .

Question 15

In order to find the correlation coefficient, the estimated least-squares regression line is always found first.

Question 16

It is never possible for $\sum(\hat{y}-\bar{y})^{2}$ to exceed $\sum(y-\bar{y})^{2}$ .

Question 17

A high correlation between two variables will not prove that one variable causes the other to occur.

Question 18

The hypothesis of no correlation between two variables can always be rejected if $r$ is greater than zero.

Question 19

The value of $z$ is a natural logarithm.

Question 20

If the correlation between $x$ and $y$ is not zero, then the variability about the estimated regression line will be less than the total variability in $y$ .

Question 21

The value $r^{2}$ can assume any value between __________.

Question 22

The sample coefficient $r$ is an estimate of __________.

Question 23

If the multiple correlation coefficient is 0.80 , then the proportion of total variation in $y$ that can be attributed to the $x$ 's is __________.

Question 24

The least squares equation $y=3+4 x_{1}+5 x_{2}$ is considered to be a better predictor of $y$ than the least squares equation $y=7+5 x_{3}$ , if the multiple correlation coefficient is __________ than the absolute value of the correlation coefficient involving $x_{1}$ and $x_{3}$ .

Question 25

If a regression line is horizontal, then the correlation between the two variables is __________.

Question 26

If all data points lie on a regression line having a nonzero slope, then $r^{2}$ equals __________.

Question 27

If all data points lie on the regression line, then the standard error of estimate is __________.

Question 28

The proportion of total variation which is unexplained can be denoted in symbols by __________.