Q: A business fits the regression model $$ \hat{y} = 12.47 + 1.74x $$ for weekly sales ($$y) $$and advertising spend ($$x in $$thousands of dollars). What is the predicted weekly sales if the advertising spend is $$20$$ thousand dollars?

$$ \hat{y} = 12.47 + 1.74 \times 20 = 47.27 $$

Question 1

Which of the following best describes the purpose of a scatterplot in statistics?

Accepted Answer

To visualize the relationship between two quantitative variables

Question 2

Given the formula for Pearson's correlation coefficient $$r$$:

$$ r = \frac{SS_{xy}}{\sqrt{SS$$_{xx}$$SS_{yy}}} $$

What does a value of $$r = -0.85$$ indicate about the relationship between two variables?

Accepted Answer

A strong negative linear association

Question 3

Suppose you have the following sums for a dataset:

$$ SS_{xy} = 50 $$, $$ SS_{xx} = 100 $$, $$ SS_{yy} = 200 $$

Calculate the correlation coefficient $$r.$$

Accepted Answer

$$ r = 0.5 $$

Question 4

Which of the following statements about the correlation coefficient $$r is $$TRUE?

Accepted Answer

$$r$$ can only take values between $$-1$$ and $$1$$

Question 5

A study finds a correlation of $$r = 0.93$$ between age and systolic blood pressure in adults. Which of the following is the best interpretation?

Accepted Answer

There is a strong positive linear association between age and systolic blood pressure

Question 6

Which of the following is NOT a property of Pearson's correlation coefficient?

Accepted Answer

It can be used for categorical variables

Question 7

Given the regression equation $$ \hat{y} = 2.5 + 0.8x $$, what does the slope $$0.8$$ represent in a real-world context where $$x is $$hours studied and $$y is $$exam score?

Accepted Answer

For each additional hour studied, the predicted exam score increases by 0.8 points

Question 8

Why do we square the residuals when calculating the least squares regression line?

Accepted Answer

To ensure all errors are positive and to penalize larger errors more heavily

Question 9

Suppose the sum of squared errors (SSE) for a regression line is $$0.2727. $$What does this value represent?

Accepted Answer

The total squared deviation of observed values from the regression line

Question 10

Which of the following best describes the intercept ($$b_0) in $$the regression equation $$ \hat{y} = b_0$ + b_1$ x $$?

Accepted Answer

The predicted value of $$y$$ when $$x = 0$$

Question 11

A business fits the regression model $$ \hat{y} = 12.47 + 1.74x $$ for weekly sales ($$y) $$and advertising spend ($$x in $$thousands of dollars). What is the predicted weekly sales if the advertising spend is $$20$$ thousand dollars?

Accepted Answer

$$ \hat{y} = 12.47 + 1.74 	imes 20 = 47.27 $$

Question 12

Which of the following is a correct interpretation of a residual in regression analysis?

Accepted Answer

The difference between the observed value and the predicted value

Question 13

If the correlation coefficient $$r is $$close to zero, what does this imply about the relationship between $$x$$ and $$y$$?

Accepted Answer

There is no linear relationship between $$x$$ and $$y$$

Question 14

Which of the following is the formula for the least squares estimate of the slope $$b_1 in $$simple linear regression?

Accepted Answer

$$ b_1$ = \frac{SS_{xy}}{SS_{xx}} $$

Question 15

Which of the following scenarios is most likely to result in a misleading correlation coefficient?

Accepted Answer

Presence of outliers in the data

Question 16

In the context of regression, what does the method of least squares accomplish?

Accepted Answer

It finds the line that minimizes the sum of squared residuals

Question 17

Which of the following is a limitation of Pearson's correlation coefficient?

Accepted Answer

It only measures linear relationships

Question 18

Suppose you observe a scatterplot with a clear upward trend and a correlation coefficient of $$r = 0.85. $$Which of the following best describes the association?

Accepted Answer

Strong positive linear association

Question 19

If $$SS_{xy} is $$zero, what can you conclude about the relationship between $$x$$ and $$y$$?

Accepted Answer

There is no linear association between $$x$$ and $$y$$

Correlation and Simple Linear Regression: Study Notes

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