Question 1

For the situation above, write the equation of the least squares line. 
A) GMAT = 228 - 92040 (SALARY)
B) SALARY $= 228 + 92040$ (GMAT)
C) SALARY $= - 92040 + 228$ (GMAT)
D) GMAT $= - 92040 + 228$ (SALARY)

Accepted Answer

The answer of For the situation above, write the equation...

Question 2

For the situation above, give a practical interpretation of $r = .81$.
A) We can predict SALARY correctly $81 \%$ of the time using GMAT in a straight-line model.
B) We estimate SALARY to increase $81 \%$ for every 1 -point increase in GMAT.
C) $81 \%$ of the sample variation in SALARY can be explained by using GMAT in a straight-line model.
D) There appears to be a positive correlation between SALARY and GMAT.

Accepted Answer

An $r$ value of 0.81 indicates a strong positive correlation between SALARY and GMAT, meaning as GMAT scores increase, SALARY tends to increase as well.

Question 3

Set up the null and alternative hypotheses for testing whether a positive linear relationship exists between SALARY and GMAT in the situation above.
A) $H _ { 0 } : \beta _ { 1 } = 228$ vs. $H _ { \mathrm { a } } : \beta _ { 1 } > 228$
B) $H _ { 0 } : \beta _ { 1 } > 0$ vs. $H _ { \mathrm { a } } : \beta _ { 1 } < 0$
C) $H _ { 0 } : \beta _ { 1 } = 0$ vs. $H _ { \mathrm { a } } : \beta _ { 1 } > 0$
D) $H _ { 0 } : \beta 1 = 0$ vs. $H _ { \mathrm { a } } : \beta _ { 1 } \neq 0$

Accepted Answer

To test for a positive linear relationship, the null hypothesis should state there is no relationship ($\beta_1 = 0$), and the alternative should suggest a positive relationship ($\beta_1 > 0$).

Question 4

For the situation above, give a practical interpretation of $r ^ { 2 } = .66$.
A) We estimate SALARY to increase $\$ .66$ for every 1-point increase in GMAT.
B) We can predict SALARY correctly $66 \%$ of the time using GMAT in a straight-line model.
C) We expect to predict SALARY to within $2 ( \sqrt { .66 } )$ of its true value using GMAT in a straight-line model.
D) $66 \%$ of the sample variation in SALARY can be explained by using GMAT in a straight -line model.

Accepted Answer

$r^2 = .66$ indicates that 66% of the variance in the dependent variable (SALARY) can be explained by the independent variable (GMAT) in a linear regression model.

Question 5

For the situation above, give a practical interpretation of $\hat { \beta } _ { 0 } = - 92040$.
A) We estimate SALARY to decrease $\$ 92,040$ for every 1 -point increase in GMAT.
B) The value has no practical interpretation since a GMAT of 0 is nonsensical and outside the range of the sample data.
C) We estimate the base SALARY of graduates of a top business school to be $- \$ 92,040$.
D) We expect to predict SALARY to within $2 ( 92040 ) = \$ 184,080$ of its true value using GMAT in a straight-line model.

Accepted Answer

The value of $\hat { \beta } _ { 0 }$ represents the intercept of a regression line, which in this context would be the estimated salary when the GMAT score is 0. Since a GMAT score of 0 is unrealistic and outside the range of possible scores, this value does not have a practical interpretation in the context of predicting salaries based on GMAT scores.

Question 6

For the situation above, which of the following is not an assumption required for the simple linear regression analysis to be valid?
A)The errors of predicting SALARY are normally distributed.
B)The errors of predicting SALARY have a variance that is constant for any given value of GMAT.
C)SALARY is independent of GMAT.
D)The errors of predicting SALARY have a mean of 0.

Accepted Answer

For simple linear regression analysis to be valid, it is assumed that the dependent variable (SALARY) is related to the independent variable (GMAT), not independent of it. Assumptions A, B, and D are standard for regression analysis, focusing on the distribution, variance, and mean of errors, respectively.

Question 7

A $95 \%$ prediction interval for SALARY when GMAT $= 600$ is $( \$ 37,915 , \$ 51,948 )$. Interpret this interval for the situation above.

A) We are $95 \%$ confident that the mean SALARY of all top business school graduates with GMATs of 600 will fall between $\$ 37,915$ and $\$ 51,984$.
B) We are $95 \%$ confident that the SALARY of a top business school graduate with a GMAT of 600 will fall between $\$ 37,915$ and $\$ 51,984$.
C) We are $95 \%$ confident that the SALARY of a top business school graduate will fall between $\$ 37,915$ and $\$ 51,984$.
D) We are $95 \%$ confident that the increase in SALARY for a 600-point increase in GMAT will fall between $\$ 37,915$ and $\$ 51,984$.

Accepted Answer

A prediction interval provides an estimate of where an individual observation's value (in this case, SALARY) will fall, given a certain level of confidence. Choice B correctly interprets the prediction interval for an individual graduate's salary, not the mean salary (A), not for all graduates regardless of GMAT score (C), and not for the increase in salary due to GMAT score increase (D).

Question 8

For the situation above, give a practical interpretation of $\mathrm { s } = 3213$.
A) We estimate SALARY to increase $\$ 3,213$ for every 1-point increase in GMAT.
B) We expect to predict SALARY to within $2 ( 3213 ) = \$ 6,426$ of its true value using GMAT in a straight-line model.
C) Our predicted value of SALARY will equal $2 ( 3213 ) = \$ 6,426$ for any value of GMAT.
D) We expect the predicted SALARY to deviate from actual SALARY by at least $2 ( 3213 ) = \$ 6,426$ using GMAT in a straight-line model.

Accepted Answer

The interpretation of $s = 3213$ in a statistical context typically refers to the standard error of the estimate, which measures the average distance that the observed values fall from the regression line. Therefore, $2s$ gives a range around the predicted value within which we expect actual values to fall, assuming a normal distribution of errors. This is a common way to express the accuracy of predictions in regression analysis.

Question 9

For the situation above, give a practical interpretation of $\hat { \beta } _ { 1 } = 228$.
A) We expect to predict SALARY to within $2 ( 228 ) = \$ 456$ of its true value using GMAT in a straight-line model.
B) We estimate GMAT to increase 228 points for every $\$ 1$ increase in SALARY.
C) We estimate SALARY to increase $\$ 228$ for every 1-point increase in GMAT.
D) The value has no practical interpretation since a GMAT of 228 is nonsensical and outside the range of the sample data.

Accepted Answer

The coefficient $\hat { \beta } _ { 1 } = 228$ in a regression model indicates the expected change in the dependent variable (SALARY) for a one-unit increase in the independent variable (GMAT). Therefore, for every 1-point increase in GMAT score, the SALARY is expected to increase by $228.

Question 10

For the situation above, write the equation of the probabilistic model of interest. 
A) GMAT $= \beta _ { 0 } + \beta _ { 1 }$ (SALARY)
B) Salary $= \beta _ { 0 } + \beta _ { 1 } ($ GMAT $) + \epsilon$
C) Salary $= \beta _ { 0 } + \beta _ { 1 } ($ GMAT $)$
D) GMAT $= \beta _ { 0 } + \beta _ { 1 } ($ SALARY $) + \epsilon$

Accepted Answer

The correct equation for a probabilistic model predicting Salary based on GMAT scores includes an error term ($\epsilon$) to account for randomness or unexplained variation, making option B the correct choice.

Question 11

A breeder of Thoroughbred horses wishes to model the relationship between the gestation
period and the length of life of a horse. The breeder believes that the two variables may
follow a linear trend. The information in the table was supplied to the breeder from
various thoroughbred stables across the state. $\begin{array}{cccccc}
\text { Horse } & \text { Gestation } & \text { Life } & \text { Horse } & \text { Gestation } & \text { Life } \
& \text { period } & \text { Length } & & \text { period } & \text { Length } \
& x \text { (days) } & y \text { (years) } & & x \text { (days) } & y \text { (years) } \
1 & 416 & 24 & 5 & 356 & 22 \
2 & 279 & 25.5 & 6 & 403 & 23.5 \
3 & 298 & 20 & 7 & 265 & 21 \
4 & 307 & 21.5 & & &
\end{array}$
Summary statistics yield $S S _ { x x } = 21,752 , S S _ { x y } = 236.5 , S S _ { y y } = 22 , \bar { x } = 332$, and $\bar { y } = 22.5$. Find a $95 \%$ prediction interval for the length of life of a horse that had a gestation period of 300 days. Use $s = 2$ as an estimate of $\sigma$ and use $\hat { y } = 18.89 + .01087 x$.

Accepted Answer

The answer of A breeder of Thoroughbred horses wishes to...

Question 12

For the situation above, give a practical interpretation of $t = 6.67$.
A) There is evidence (at $\alpha = .05$ ) to indicate that $\beta _ { 1 } = 0$.
B) There is evidence (at $\alpha = .05$ ) of at least a positive linear relationship between SALARY and GMAT.
C) We estimate SALARY to increase $\$ 6.67$ for every 1 -point increase in GMAT.
D) Only $6.67 \%$ of the sample variation in SALARY can be explained by using GMAT in a straight-line model.

Accepted Answer

The t-value of 6.67 suggests a statistically significant positive linear relationship between SALARY and GMAT at the 0.05 significance level.

Question 13

Consider the following pairs of observations: $\begin{array}{|l|l|l|l|l|l|}
\hline x & 2 & 0 & 3 & 3 & 5 \
\hline y & 1 & 3 & 4 & 6 & 7 \
\hline
\end{array}$

a. Construct a scattergram for the data.
b. Find the least squares line, and plot it on your scattergram.
c. Find a $99 \%$ confidence interval for the mean value of $y$ when $x = 1$.
d. Find a $99 \%$ prediction interval for a new value of $y$ when $x = 1$.

Accepted Answer

The answer of Consider the following pairs of observations: \(\begin{array}{|l|l|l|l|l|l|}
\hline...

Quiz 11: Simple Linear Regression

For the situation above, write the equation of the least squares line.

For the situation above, give a practical interpretation of $r = .81$ .

Set up the null and alternative hypotheses for testing whether a positive linear relationship exists between SALARY and GMAT in the situation above.

For the situation above, give a practical interpretation of $r ^ { 2 } = .66$ .

For the situation above, give a practical interpretation of $\hat { \beta } _ { 0 } = - 92040$ .

For the situation above, which of the following is not an assumption required for the simple linear regression analysis to be valid?

A $95 \%$ prediction interval for SALARY when GMAT $= 600$ is $( \$ 37,915 , \$ 51,948 )$ . Interpret this interval for the situation above.

For the situation above, give a practical interpretation of $\mathrm { s } = 3213$ .

For the situation above, give a practical interpretation of $\hat { \beta } _ { 1 } = 228$ .

For the situation above, write the equation of the probabilistic model of interest.

For the situation above, give a practical interpretation of $t = 6.67$ .

Quiz 11: Simple Linear Regression

For the situation above, write the equation of the least squares line.

For the situation above, give a practical interpretation of r=.81r = .81r=.81 .

Set up the null and alternative hypotheses for testing whether a positive linear relationship exists between SALARY and GMAT in the situation above.

For the situation above, give a practical interpretation of r2=.66r ^ { 2 } = .66r2=.66 .

For the situation above, give a practical interpretation of β^0=−92040\hat { \beta } _ { 0 } = - 92040β^​0​=−92040 .

For the situation above, which of the following is not an assumption required for the simple linear regression analysis to be valid?

A 95%95 \%95% prediction interval for SALARY when GMAT =600= 600=600 is ($37,915,$51,948) ( \$ 37,915 , \$ 51,948 ) ($37,915,$51,948) . Interpret this interval for the situation above.

For the situation above, give a practical interpretation of s=3213\mathrm { s } = 3213s=3213 .

For the situation above, give a practical interpretation of β^1=228\hat { \beta } _ { 1 } = 228β^​1​=228 .