Question 1

A simple linear regression equation is given by $\hat { y }$ = 5.25 + 3.8x. The point estimate of $y$ when $x$ = 4 is 20.45.

Accepted Answer

A) True 
 B)False

Question 2

Given the least squares regression line $\hat { y }$ = 3.52 - 1.27x, and a coefficient of determination of 0.81, the coefficient of correlation is:

Accepted Answer

A) -0.85. 
B) 0.85. 
C) -0.90. 
D) 0.90. 
A) -0.85. 
B) 0.85. 
C) -0.90. 
D) 0.90.

Question 3

In a simple linear regression model, testing whether the slope, $\beta _ { 1 }$ , of the population regression line is zero is the same as testing whether the population coefficient of correlation, $\rho$ , equals zero.

Accepted Answer

A) True 
 B)False

Question 4

The residual $r _ { i }$ is defined as the difference between the actual value $y _ { i }$ and the estimated value $\hat { y } _ { i }$ .

Accepted Answer

A) True 
 B)False

Question 5

The confidence interval estimate of the expected value of y will be narrower than the prediction interval for the same given value of x and confidence level. This is because there is less error in estimating a mean value than in predicting an individual value.

Accepted Answer

A) True 
 B)False

Question 6

Would the test of the significance of the overall equation have the same conclusion as the test of significance of the slope in a simple linear regression model of weekly sales in a fast food restaurant on number of vouchers printed in the local newspaper? Explain.

Accepted Answer

Yes; because we have a simple

Question 7

An ardent fan of television game shows has observed that, in general, the more educated the contestant, the less money he or she wins. To test her belief, she gathers data about the last eight winners of her favourite game show. She records their winnings in dollars and their years of education and performs a simple linear regression, with the Excel output provided. $$\begin{array} { | c | c | c | } 
\hline \text { Contestant } & \begin{array} { c } 
\text { Years of } \
\text { education }
\end{array} & \text { Winnings } \
\hline 1 & 11 & 750 \
\hline 2 & 15 & 400 \
\hline 3 & 12 & 600 \
\hline 4 & 16 & 350 \
\hline 5 & 11 & 800 \
\hline 6 & 16 & 300 \
\hline 7 & 13 & 650 \
\hline 8 & 14 & 400 \
\hline
\end{array}$$ \(\begin{array}{|l|r|c|c|c|c|c|}
\hline \begin{array}{l}
\text { SUMMARY } \
\text { OUTPUT }
\end{array} & & \
\hline \begin{array}{c}
\text { Regression } \
\text { Statistios }
\end{array} & & \
\hline \text { Multiple R } & 0.9583798 & \
\hline \text { R Square } & 0.9184918 & \
\hline \text { Adjusted R Square } & 0.9049071 & \
\hline \text { Standard Error } & 59.395099 & \
\hline \text { Obseruations } & 8 & \
\hline & & \
\hline \text { ANOVA } & & & & & & \
\hline & \text { df } & S S & M S & F & \begin{array}{c}
\text { Significance } \
F
\end{array} & \
\hline \text { Regression } & 1 & 238520.8333 & 238520.8 & 67.6122047 & 0.00017466 & \
\hline \text { Residual } & 6 & 21166.66667 & 3527.778 & & & \
\hline \text { Total } & 7 & 259687.5 & & & & \\hline & & \
\hline & \text { coefficients } & \begin{array}{c}
\text { Standard } \
\text { Error }
\end{array} & \text { tstat } & \text { Pvalue } & \text { Lower 95\% } & \begin{array}{c}
\text { Upper } \
95 \%
\end{array} \
\hline \text { Intercept } & 1735 & 147.8926037 & 11.73149 & 2.3148 E-05 & 1373.11984 & 2096.8802 \
\hline \text { Years of education } & -89.16667 & 10.84401183 & 8.222664 & 0.00017466 & -115.70101 & -62.63233 \
\hline
\end{array}\) a. Determine the least squares regression line.
b. Interpret the value of the slope of the regression line.
c. Determine the standard error of estimate, and describe what this statistic tells you about the regression line.
d. Interpret the coefficient of correlation

Accepted Answer

a. @#IMG-DLM& = 1735 - 89.1667x.
b. For each addit

Question 8

A statistician investigating the relationship between the amount of precipitation (in inches) and the number of car accidents gathered data for 10 randomly selected days. The results are presented below. $$\begin{array} { | c | c | c | } 
\hline \text { Day } & \text { Precipitation } & \text { Number of accidents } \
\hline 1 & 0.05 & 5 \
\hline 2 & 0.12 & 6 \
\hline 3 & 0.05 & 2 \
\hline 4 & 0.08 & 4 \
\hline 5 & 0.10 & 8 \
\hline 6 & 0.35 & 14 \
\hline 7 & 0.15 & 7 \
\hline 8 & 0.30 & 13 \
\hline 9 & 0.10 & 7 \
\hline 10 & 0.20 & 10 \
\hline
\end{array}$$ Do these data allow us to conclude at the 10% significance level that the amount of precipitation and the number of accidents are linearly related?

Accepted Answer

H₀: β₁ = 0 H_A: β₁ ≠ 0 Rejection re

Question 9

Given that cov(x,y) = 8.5, $s _ { y } ^ { 2 }$ = 8 and $S _ { x } ^ { 2 }$ = 10, the value of the coefficient of determination is 0.95.

Accepted Answer

A) True 
 B)False

Question 10

The value of the sum of squares for regression, SSR, can never be larger than the value of sum of squares for error, SSE.

Accepted Answer

A) True 
 B)False

Question 11

A professor of economics wants to study the relationship between income y (in $1000s) and education x (in years). A random sample of eight individuals is taken and the results are shown below. $$\begin{array} { | l | c | c | c | c | c | c | c | c | } 
\hline \text { Education } & 16 & 11 & 15 & 8 & 12 & 10 & 13 & 14 \
\hline \text { Income } & 58 & 40 & 55 & 35 & 43 & 41 & 52 & 49 \
\hline
\end{array}$$ Determine the coefficient of determination, and discuss what its value tells you about the two variables.

Accepted Answer

@#IMG-DLM& 0.9223, which means that 92.0

Question 12

A regression analysis between sales (in $1000) and advertising (in $) yielded the least squares line $\hat { y }$ = 80 000 + 5x. This implies that an:

Accepted Answer

A) increase of $1 in advertising is expected to result in an increase of $5 in sales. 
B) increase $5 in advertising is expected to result in an increase of $5000 in sales. 
C) increase of $1 in advertising is expected to result in an increase of $80 005 in sales. 
D) increase of $1 in advertising is expected to result in an increase of $5000 in sales. 
A) increase of $1 in advertising is expected to result in an increase of $5 in sales. 
B) increase $5 in advertising is expected to result in an increase of $5000 in sales. 
C) increase of $1 in advertising is expected to result in an increase of $80 005 in sales. 
D) increase of $1 in advertising is expected to result in an increase of $5000 in sales.

Question 13

The value of the sum of squares for regression, SSR, can never be smaller than 0.0.

Accepted Answer

A) True 
 B)False

Question 14

The manager of a fast food restaurant wants to determine how sales in a given week are related to the number of discount vouchers (#) printed in the local newspaper during the week. The number of vouchers and sales ($000s) from 10 randomly selected weeks is given below with Excel regression output. $$\begin{array} { | c | c | } 
\hline \text { Number of vouchers } & \text { Sales } \
\hline 4 & 12.8 \
\hline 7 & 15.4 \
\hline 5 & 13.9 \
\hline 3 & 11.2 \
\hline 19 & 18.7 \
\hline 10 & 17.9 \
\hline 8 & 16.8 \
\hline 6 & 15.9 \
\hline 3 & 11.5 \
\hline 5 & 13.9 \
\hline
\end{array}$$ \(\begin{array}{|l|c|c|c|c|c|l|}
\hline \text { SUMMARY OUTPUT } & \
\hline & \
\hline \text { Regression Statistics } & \
\hline \text { Multiple R } & 0.8524 \
\hline \text { R Square } & 0.7267 \
\hline \text { Adjusted R Square } & 0.6925 \
\hline \text { Standard Error } & 1.4301 \
\hline \text { Observations } & 10 \\hline & \
\hline \text { ANOvA } & & & & & & \
\hline & d f & S S & M S & F & \text { Significance } F & \
\hline \text { Regression } & 1 & 43.4982 & 43.4982 & 21.2682 & 0.0017 & \
\hline \text { Residual } & 8 & 16.3618 & 2.0452 & & & \
\hline \text { Total } & 9 & 59.8600 & & & & \\hline\
\hline & \text { Coeficients } & \begin{array}{c}
\text { Standard } \
\text { Error }
\end{array} & t \text { Stat } & \text { P-value } & \text { Lower } 95 \% & \begin{array}{c}
\text { Upper } \
95 \%
\end{array} \
\hline \text { Intercept } & 11.5676 & 0.8341 & 13.8679 & 0.0000 & 9.6441 & 13.4912 \
\hline \text { Number of vouchers } & 0.4618 & 0.1001 & 4.6117 & 0.0017 & 0.2309 & 0.6927 \
\hline
\end{array}\) Interpret the coefficient of correlation.

Accepted Answer

r = 0.8524
There is a positive, strong,

Question 15

In regression analysis, if the coefficient of correlation is -1.0, then:

Accepted Answer

A) the sum of squares for error is -1.0. 
B) the sum of squares for regression is 1.0. 
C) the sum of squares for error and sum of squares for regression are equal. 
D) the sum of squares for regression and total variation in y are equal. 
A) the sum of squares for error is -1.0. 
B) the sum of squares for regression is 1.0. 
C) the sum of squares for error and sum of squares for regression are equal. 
D) the sum of squares for regression and total variation in y are equal.

Question 16

The least squares method requires that the variance $\sigma _ { \varepsilon } ^ { 2 }$ of the error variable ? is a constant no matter what the value of x is. When this requirement is violated, the condition is called:

Accepted Answer

A) multicollinearity. 
B) heteroscedasticity. 
C) homoscedasticity. 
D) autocorrelation. 
A) multicollinearity. 
B) heteroscedasticity. 
C) homoscedasticity. 
D) autocorrelation.

Question 17

In testing the hypotheses: H0: F1F1F1S1F1F1F10s = 0
HA: F1F1F1S1F1F1F10s F1F1F1S1F1F1F10 0
The Spearman rank correlation coefficient in a sample of 50 observations is 0.389. The value of the test statistic is:

Accepted Answer

A) 2.75. 
B) 18.178. 
C) 2.723. 
D) 17.995. 
A) 2.75. 
B) 18.178. 
C) 2.723. 
D) 17.995.

Question 18

When the actual values y of a dependent variable and the corresponding predicted values $\hat { y }$ are the same, the standard error of the estimate will be 1.0.

Accepted Answer

A) True 
 B)False

Question 19

An outlier is an observation that is unusually small or unusually large.

Accepted Answer

A) True 
 B)False

Question 20

Given the following linear regression model of Weekly sales ($000's) in a fast food restaurant against number of vouchers printed in the local newspaper, interpret the intercept. Does this make sense?
Estimated Sales = 11.5676 + 0.4618.Vouchers

Accepted Answer

We estimate, if there were no vouchers p

A simple linear regression equation is given by $\hat { y }$ = 5.25 + 3.8x. The point estimate of $y$ when $x$ = 4 is 20.45.

Given the least squares regression line $\hat { y }$ = 3.52 - 1.27x, and a coefficient of determination of 0.81, the coefficient of correlation is:

In a simple linear regression model, testing whether the slope, $\beta _ { 1 }$ , of the population regression line is zero is the same as testing whether the population coefficient of correlation, $\rho$ , equals zero.

The residual $r _ { i }$ is defined as the difference between the actual value $y _ { i }$ and the estimated value $\hat { y } _ { i }$ .

The confidence interval estimate of the expected value of y will be narrower than the prediction interval for the same given value of x and confidence level. This is because there is less error in estimating a mean value than in predicting an individual value.

Would the test of the significance of the overall equation have the same conclusion as the test of significance of the slope in a simple linear regression model of weekly sales in a fast food restaurant on number of vouchers printed in the local newspaper? Explain.

Given that cov(x,y) = 8.5, $s _ { y } ^ { 2 }$ = 8 and $S _ { x } ^ { 2 }$ = 10, the value of the coefficient of determination is 0.95.

The value of the sum of squares for regression, SSR, can never be larger than the value of sum of squares for error, SSE.

A regression analysis between sales (in $1000) and advertising (in $) yielded the least squares line $\hat { y }$ = 80 000 + 5x. This implies that an:

The value of the sum of squares for regression, SSR, can never be smaller than 0.0.

In regression analysis, if the coefficient of correlation is -1.0, then:

The least squares method requires that the variance $\sigma _ { \varepsilon } ^ { 2 }$ of the error variable ? is a constant no matter what the value of x is. When this requirement is violated, the condition is called:

In testing the hypotheses: H0: s = 0 HA: s  0 The Spearman rank correlation coefficient in a sample of 50 observations is 0.389. The value of the test statistic is:

When the actual values y of a dependent variable and the corresponding predicted values $\hat { y }$ are the same, the standard error of the estimate will be 1.0.

An outlier is an observation that is unusually small or unusually large.

Given the following linear regression model of Weekly sales ($000's) in a fast food restaurant against number of vouchers printed in the local newspaper, interpret the intercept. Does this make sense? Estimated Sales = 11.5676 + 0.4618.Vouchers

What Is Statistics

Types of Data, Data Collection and Sampling

Graphical Descriptive Techniques Nominal Data

Graphical Descriptive Techniques Numerical Data

Numerical Descriptive Measures

Probability

Random Variables and Discrete Probability Distributions

Continuous Probability Distributions

Statistical Inference and Sampling Distributions

Estimation: Describing a Single Population

Estimation: Comparing Two Populations

Hypothesis Testing: Describing a Single Population

Hypothesis Testing: Comparing Two Populations

Additional Tests for Nominal Data: Chi-Squared Tests

Multiple Regression

Time-Series Analysis and Forecasting

Index Numbers

Filters

Exam 15: Simple Linear Regression and Correlation

A simple linear regression equation is given by y^\hat { y }y^​ = 5.25 + 3.8x. The point estimate of yyy when xxx = 4 is 20.45.

Given the least squares regression line y^\hat { y }y^​ = 3.52 - 1.27x, and a coefficient of determination of 0.81, the coefficient of correlation is:

In a simple linear regression model, testing whether the slope, β1\beta _ { 1 }β1​ , of the population regression line is zero is the same as testing whether the population coefficient of correlation, ρ\rhoρ , equals zero.

The residual rir _ { i }ri​ is defined as the difference between the actual value yiy _ { i }yi​ and the estimated value y^i\hat { y } _ { i }y^​i​ .

The confidence interval estimate of the expected value of y will be narrower than the prediction interval for the same given value of x and confidence level. This is because there is less error in estimating a mean value than in predicting an individual value.

Would the test of the significance of the overall equation have the same conclusion as the test of significance of the slope in a simple linear regression model of weekly sales in a fast food restaurant on number of vouchers printed in the local newspaper? Explain.

Given that cov(x,y) = 8.5, sy2s _ { y } ^ { 2 }sy2​ = 8 and Sx2S _ { x } ^ { 2 }Sx2​ = 10, the value of the coefficient of determination is 0.95.

The value of the sum of squares for regression, SSR, can never be larger than the value of sum of squares for error, SSE.

A regression analysis between sales (in $1000) and advertising (in $) yielded the least squares line y^\hat { y }y^​ = 80 000 + 5x. This implies that an:

The value of the sum of squares for regression, SSR, can never be smaller than 0.0.

In regression analysis, if the coefficient of correlation is -1.0, then:

The least squares method requires that the variance σε2\sigma _ { \varepsilon } ^ { 2 }σε2​ of the error variable ? is a constant no matter what the value of x is. When this requirement is violated, the condition is called:

In testing the hypotheses: H0: s = 0 HA: s  0 The Spearman rank correlation coefficient in a sample of 50 observations is 0.389. The value of the test statistic is:

When the actual values y of a dependent variable and the corresponding predicted values y^\hat { y }y^​ are the same, the standard error of the estimate will be 1.0.

An outlier is an observation that is unusually small or unusually large.

Given the following linear regression model of Weekly sales ($000's) in a fast food restaurant against number of vouchers printed in the local newspaper, interpret the intercept. Does this make sense? Estimated Sales = 11.5676 + 0.4618.Vouchers

What Is Statistics

Types of Data, Data Collection and Sampling

Graphical Descriptive Techniques Nominal Data

Graphical Descriptive Techniques Numerical Data

Numerical Descriptive Measures

Probability

Random Variables and Discrete Probability Distributions

Continuous Probability Distributions

Statistical Inference and Sampling Distributions

Estimation: Describing a Single Population

Estimation: Comparing Two Populations

Hypothesis Testing: Describing a Single Population

Hypothesis Testing: Comparing Two Populations

Additional Tests for Nominal Data: Chi-Squared Tests

Multiple Regression

Time-Series Analysis and Forecasting

Index Numbers

Filters

A simple linear regression equation is given by $\hat { y }$ = 5.25 + 3.8x. The point estimate of $y$ when $x$ = 4 is 20.45.

Given the least squares regression line $\hat { y }$ = 3.52 - 1.27x, and a coefficient of determination of 0.81, the coefficient of correlation is:

In a simple linear regression model, testing whether the slope, $\beta _ { 1 }$ , of the population regression line is zero is the same as testing whether the population coefficient of correlation, $\rho$ , equals zero.

The residual $r _ { i }$ is defined as the difference between the actual value $y _ { i }$ and the estimated value $\hat { y } _ { i }$ .

Given that cov(x,y) = 8.5, $s _ { y } ^ { 2 }$ = 8 and $S _ { x } ^ { 2 }$ = 10, the value of the coefficient of determination is 0.95.

A regression analysis between sales (in $1000) and advertising (in $) yielded the least squares line $\hat { y }$ = 80 000 + 5x. This implies that an:

The least squares method requires that the variance $\sigma _ { \varepsilon } ^ { 2 }$ of the error variable ? is a constant no matter what the value of x is. When this requirement is violated, the condition is called:

When the actual values y of a dependent variable and the corresponding predicted values $\hat { y }$ are the same, the standard error of the estimate will be 1.0.