Deck 12: Simple Regression

Question 1

If R² = .36 in the model Sales = 268 + 7.37 Ads with n = 50, the two-tailed test for correlation at α = .05 would say that there is a significant correlation between Sales and Ads.

Answer 1

True

Question 2

The least squares regression line is obtained when the sum of the squared residuals is minimized.

Answer 2

True

Question 3

A common source of spurious correlation between X and Y is when a third unspecified variable Z affects both X and Y.

Answer 3

True

Question 4

In a simple regression, if the coefficient for X is positive and significantly different from zero, then an increase in X is associated with an increase in the mean (i.e., the expected value) of Y.

Question 5

Pearson's correlation coefficient (r) requires that both variables be interval or ratio data.

Question 6

The fitted intercept in a regression has little meaning if no data values near X = 0 have been observed.

Question 7

The least squares regression line gives unbiased estimates of β₀ and β₁.

Question 8

If SSR is 1800 and SSE is 200, then R² is .90.

Question 9

A sample correlation r = .40 indicates a stronger linear relationship than r = -.60.

Question 10

A scatter plot is used to visualize the association (or lack of association) between two quantitative variables.

Question 11

In a simple regression, the correlation coefficient r is the square root of R².

Question 12

If r = .55 and n = 16, then the correlation is significant at α = .05 in a two-tailed test.

Question 13

The correlation coefficient r always has the same sign as b₁ in Y = b₀ + b₁X.

Question 14

If R² = .36 in the model Sales = 268 + 7.37 Ads, then Ads explains 36 percent of the variation in Sales.

Question 15

In the model Sales = 268 + 7.37 Ads (both variables in dollars) an additional $1 spent on ads will increase sales by 7.37 percent.

Question 16

When using the least squares method, the column of residuals always sums to zero.

Question 17

The width of a prediction interval for an individual value of Y is less than standard error s_e.

Question 18

The correlation coefficient r measures the strength of the linear relationship between two variables.

Question 19

The ordinary least squares regression line always passes through the point $( \bar { x } , \bar { y } )$

Question 20

In least squares regression, the residuals e₁, e₂,…, e_n will always have a zero mean.

Question 21

Studentized (or standardized) residuals permit us to detect cases where the regression predicts poorly.

Question 22

Confidence intervals for predicted Y are less precise when the residuals are very small.

Question 23

In simple linear regression, the p-value of the slope will always equal the p-value of the F statistic.

Question 24

For a regression with 200 observations, we expect that about 10 residuals will exceed two standard errors.

Question 25

An observation with high leverage will have a large residual (usually an outlier).

Question 26

When X is farther from its mean, the prediction interval and confidence interval for Y become wider.

Question 27

In simple linear regression, the coefficient of determination (R²) is estimated from sums of squares in the ANOVA table.

Question 28

The larger the absolute value of the t statistic of the slope in a simple linear regression, the stronger the linear relationship exists between X and Y.

Question 29

If you have a strong outlier in the residuals, it may represent a different causal system.

Question 30

In linear regression between two variables, a significant relationship exists when the p-value of the t test statistic for the slope is greater than α.

Question 31

Cause-and-effect direction between X and Y may be determined by running the regression twice and seeing whether Y = β₀ + β₁X or X = β₁ + β₀Y has the larger R².

Question 32

If SSE is near zero in a regression, the statistician will conclude that the proposed model probably has too poor a fit to be useful.

Question 33

A poor prediction (large residual) indicates an observation with high leverage.

Question 34

Ill-conditioned refers to a variable whose units are too large or too small .

Question 35

A prediction interval for Y is narrower than the corresponding confidence interval for the mean of Y.

Question 36

A negative correlation between two variables X and Y usually yields a negative p-value for r.

Question 37

The total sum of squares (SST) will never exceed the regression sum of squares (SSR).

Question 38

"High leverage" would refer to a data point that is poorly predicted by the model (large residual).

Question 39

The ordinary least squares method of estimation minimizes the estimated slope and intercept.

Question 40

The ordinary least squares method ensures that the residuals will be normally distributed.

Question 41

A prediction interval for Y is widest when X is near its mean.

Question 42

A predictor that is significant in a one-tailed t-test will also be significant in a two-tailed test at the same level of significance α.

Question 43

In a simple regression, there are n - 2 degrees of freedom associated with the error sum of squares (SSE).

Question 44

Outliers can be detected by examining the standardized residuals.

Question 45

Using the least squares formulas, the regression line must pass through the origin.

Question 46

The coefficient of determination is the percentage of the total variation in the response variable Y that is explained by the predictor X.

Question 47

When the errors in a regression model are not independent, the regression model is said to have autocorrelation.

Question 48

In a simple regression, the F statistic is calculated by taking the ratio of MSR to the MSE.

Question 49

A simple decimal transformation often improves data conditioning.

Question 50

In a simple bivariate regression, F_calc = t_calc².

Question 51

Autocorrelated errors are not usually a concern for regression models using cross-sectional data.

Question 52

In correlation analysis, neither X nor Y is designated as the independent variable.

Question 53

A different confidence interval exists for the mean value of Y for each different value of X.

Question 54

Omission of a relevant predictor is a common source of model misspecification.

Question 55

A negative value for the correlation coefficient (r) implies a negative value for the slope (b₁).

Question 56

Correlation analysis primarily measures the degree of the linear relationship between X and Y.

Question 57

Two-tailed t-tests are often used because any predictor that differs significantly from zero in a two-tailed test will also be significantly greater than zero or less than zero in a one-tailed test at the same α.

Question 58

In a two-tailed test for correlation at α = .05, a sample correlation coefficient r = 0.42 with n = 25 is significantly different than zero.

Question 59

There are usually several possible regression lines that will minimize the sum of squared errors.

Question 60

High leverage for an observation indicates that X is far from its mean.

Question 61

A researcher's Excel results are shown below using Femlab (labor force participation rate among females) to try to predict Cancer (death rate per 100,000 population due to cancer) in the 50 U.S. states. $$\begin{array} { l r } \hline { \text { Regression Statistics } } \\\hline \text { Multiple R } & 0.313422848 \\\text { R Square } & 0.098233882 \\\text { Adjusted R Square } & 0.079447088 \\\text { Standard Error } & 32.07003698 \\\text { Observations } & 50 \\\hline\end{array}$$ $$\begin{array} { l r r r } \hline \text { Variable } & \text { Coefficients } & \text { Standard Error } & { t \text { Stat } } \\\hline \text { Intercept } & 343.619889 & 61.0823514 & 5.62552 \\\text { Femlab } & - 2.2833659 & 0.99855319 & - 2.28667 \\\hline\end{array}$$ Which of the following statements is not true?

A)The standard error is too high for this model to be of any predictive use.
B)The 95 percent confidence interval for the coefficient of Femlab is -4.29 to -0.28.
C)Significant correlation exists between Femlab and Cancer at ? = .05.
D)The two-tailed p-value for Femlab will be less than .05.

Question 62

Prediction intervals for Y are narrowest when:

A)the mean of X is near the mean of Y.
B)the value of X is near the mean of X.
C)the mean of X differs greatly from the mean of Y.
D)the mean of X is small. $\bar { x }$ $\bar { x }$

Question 63

A local trucking company fitted a regression to relate the travel time (days) of its shipments as a function of the distance traveled (miles). The fitted regression is Time = -7.126 + 0.0214 Distance, based on a sample of 20 shipments. The estimated standard error of the slope is 0.0053. Find the value of t_calc to test for zero slope.

A)2.46
B)5.02
C)4.04
D)3.15 $b _ { 1 } / s _ { b }$

Question 64

A news network stated that a study had found a positive correlation between the number of children a worker has and his or her earnings last year. You may conclude that:

A)people should have more children so they can get better jobs.
B)the data are erroneous because the correlation should be negative.
C)causation is in serious doubt.
D)statisticians have small families.

Question 65

A standardized residual equal to -2.205 indicates:

A)a rather poor prediction.
B)an extreme outlier in the residuals.
C)an observation with high leverage.
D)a likely data entry error.

Question 66

Which of the following is not a characteristic of the F-test in a simple regression?

A)It is a test for overall fit of the model.
B)The test statistic can never be negative.
C)It requires a table with numerator and denominator degrees of freedom.
D)The F-test gives a different p-value than the t-test.

Question 67

A local trucking company fitted a regression to relate the travel time (days) of its shipments as a function of the distance traveled (miles). The fitted regression is Time = -7.126 + .0214 Distance, based on a sample of 20 shipments. The estimated standard error of the slope is 0.0053. Find the critical value for a right-tailed test to see if the slope is positive, using α = .05.

A)2.101
B)2.552
C)1.960
D)1.734

Question 68

William used a sample of 68 large U.S. cities to estimate the relationship between Crime (annual property crimes per 100,000 persons) and Income (median annual income per capita, in dollars). His estimated regression equation was Crime = 428 + 0.050 Income. We can conclude that:

A)the slope is small so Income has no effect on Crime.
B)crime seems to create additional income in a city.
C)wealthy individuals tend to commit more crimes, on average.
D)the intercept is irrelevant since zero median income is impossible in a large city.

Question 69

The ordinary least squares (OLS) method of estimation will minimize:

A)neither the slope nor the intercept.
B)only the slope.
C)only the intercept.
D)both the slope and intercept.

Question 70

If the attendance at a baseball game is to be predicted by the equation Attendance = 16,500 - 75 Temperature, what would be the predicted attendance if Temperature is 90 degrees?

A)6,750
B)9,750
C)12,250
D)10,020

Question 71

A researcher's results are shown below using Femlab (labor force participation rate among females) to try to predict Cancer (death rate per 100,000 population due to cancer) in the 50 U.S. states. $$\begin{array} { l r } \hline { \text { Regression Statistics } } \\\hline \text { Multiple R } & 0.313422848 \\\text { R Square } & 0.098233882 \\\text { Adjusted R Square } & 0.079447088 \\\text { Standard Error } & 32.07003698 \\\text { Observations } & 50 \\\hline\end{array}$$ $$\begin{array} { l c r r } \hline \text { Variable } & \text { Coefficients } & \text { Standard Error } & { t \text { Stat } } \\\hline \text { Intercept } & 343.619889 & 61.0823514 & 5.62552 \\\text { Femlab } & - 2.2833659 & 0.99855319 & - 2.28667 \\\hline\end{array}$$ Which statement is valid regarding the relationship between Femlab and Cancer?

A)A rise in female labor participation rate will cause the cancer rate to decrease within a state.
B)This model explains about 10 percent of the variation in state cancer rates.
C)At the .05 level of significance, there isn't enough evidence to say the two variables are related.
D)If your sister starts working, the cancer rate in your state will decline.

Question 72

The variable used to predict another variable is called the:

A)response variable.
B)regression variable.
C)independent variable.
D)dependent variable.

Question 73

The standard error of the regression:

A)is based on squared deviations from the regression line.
B)may assume negative values if b₁ < 0.
C)is in squared units of the dependent variable.
D)may be cut in half to get an approximate 95 percent prediction interval.

Question 74

In a simple regression, which would suggest a significant relationship between X and Y?

A)Large p-value for the estimated slope.
B)Large t statistic for the slope.
C)Large p-value for the F statistic.
D)Small t statistic for the slope.

Question 75

Amelia used a random sample of 100 accounts receivable to estimate the relationship between Days (number of days from billing to receipt of payment) and Size (size of balance due in dollars). Her estimated regression equation was Days = 22 + 0.0047 Size with a correlation coefficient of .300. From this information we can conclude that:

A)9 percent of the variation in Days is explained by Size.
B)autocorrelation is likely to be a problem.
C)the relationship between Days and Size is significant.
D)larger accounts usually take less time to pay.

Question 76

A hypothesis test is conducted at the 5 percent level of significance to test whether the population correlation is zero. If the sample consists of 25 observations and the correlation coefficient is 0.60, then the computed test statistic would be:

A)2.071.
B)1.960.
C)3.597.
D)1.645.

Question 77

Mary used a sample of 68 large U.S. cities to estimate the relationship between Crime (annual property crimes per 100,000 persons) and Income (median annual income per capita, in dollars). Her estimated regression equation was Crime = 428 + 0.050 Income. If Income decreases by 1000, we would expect that Crime will:

A)increase by 428.
B)decrease by 50.
C)increase by 500.
D)remain unchanged.

Question 78

Using a two-tailed test at α = .05 for n = 30, we would reject the hypothesis of zero correlation if the absolute value of r exceeds:

A).2992.
B).3609.
C).0250.
D).2004.

Question 79

If n = 15 and r = .4296, the corresponding t statistic to test for zero correlation is:

A)1.715.
B)7.862.
C)2.048.
D)impossible to determine without α.

Question 80

A researcher's results are shown below using Femlab (labor force participation rate among females) to try to predict Cancer (death rate per 100,000 population due to cancer) in the 50 U.S. states. $$\begin{array} { l c r c c } \hline \text { Source of variation } & d f & { \text { SS } } & M S & F \\\hline \text { Regression } & 1 & 5377.836 & 5377.836 & 5.228879 \\\text { Residual } & 48 & 49367.389 & 1028.487 & \\\text { Total } & 49 & 54745.225 & & \\\hline\end{array}$$ What is the R² for this regression?

A).9018
B).0982
C).8395
D).1605