Q: Given $$r = 0.773$$, $$s_x = 52.23$$, $$s_y = 0.78$$, what is the slope $$b_1 of $$the least-squares regression line?

$$b_1 = 0.773$$ \times 0.78 / 52.23$$

Q: If the least-squares regression line is $$\hat{y} = 0.0116x + 5.5273$$, what is the predicted drilling time when drilling begins at 130 feet?

$$\hat{y} = 0.0116 \times 130 + 5.5273 = 7.035$$

Question 1

Which variable is considered the explanatory variable in the drilling study, and how is it represented in a scatter diagram?

Accepted Answer

Depth at which drilling begins, plotted on the horizontal axis

Question 2

Given the drilling data, what type of relationship would you expect to see in a scatter diagram between depth and time to drill?

Accepted Answer

A positive linear relation

Question 3

Which formula is used to calculate the sample linear correlation coefficient $$r$$?

Accepted Answer

$$r = \frac{\sum \left( \frac{x_i - \bar{x}}{s_x} \right) \left( \frac{y_i - \bar{y}}{s_y} \right)}{n-1}$$

Question 4

If the sample correlation coefficient $$r is 0.773 $$and the critical value for $$n=12 is 0.576$$, what can you conclude about the relationship between depth and drilling time?

Accepted Answer

There is a positive linear relation

Question 5

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

Accepted Answer

$$r is $$resistant to outliers

Question 6

What is the main difference between correlation and causation?

Accepted Answer

Correlation measures association, but causation requires evidence from a designed experiment

Question 7

Which equation represents the least-squares regression line?

Accepted Answer

$$\hat{y} = b_1$ x + b_0$

Question 8

Given $$r = 0.773$$, $$s_x = 52.23$$, $$s_y = 0.78$$, what is the slope $$b_1 of $$the least-squares regression line?

Accepted Answer

$$b_1 = 0.773$$ 	imes 0.78 / 52.23$$

Question 9

If the least-squares regression line is $$\hat{y} = 0.0116x + 5.5273$$, what is the predicted drilling time when drilling begins at 130 feet?

Accepted Answer

$$\hat{y} = 0.0116 	imes 130 + 5.5273 = 7.035$$

Question 10

What does the coefficient of determination $$R^2$$ represent in regression analysis?

Accepted Answer

The proportion of total variation in the response variable explained by the regression line

Question 11

If $$r = 0.773$$, what is the coefficient of determination $$R^2$$?

Accepted Answer

$$R^2$$ = $$0.773^2 = 0.5975$

Question 12

What is a residual in regression analysis?

Accepted Answer

The difference between the observed value of $$y$$ and the predicted value $$\hat{y}$$

Question 13

Which of the following is a sign that a linear model may NOT be appropriate based on a residual plot?

Accepted Answer

Residuals show a discernable pattern, such as a curve

Question 14

What is homoscedasticity in the context of regression analysis?

Accepted Answer

The residuals have constant variance as the explanatory variable increases

Question 15

What is an influential observation in regression analysis?

Accepted Answer

An observation that significantly affects the slope and/or y-intercept of the regression line

Question 16

In a contingency table, what is a marginal distribution?

Accepted Answer

A frequency or relative frequency distribution of either the row or column variable

Question 17

What does a conditional distribution in a contingency table show?

Accepted Answer

The relative frequency of each category of the response variable, given a specific value of the explanatory variable

Question 18

If students in the hybrid course are more likely to pass (A, B, or C) than the other two methods, what does this suggest about the association between course grade and delivery method?

Accepted Answer

There is an association between course grade and delivery method

Question 19

Why should predictions not be made outside the scope of the model using the least-squares regression line?

Accepted Answer

Because the linear relation may not continue to exist outside the observed range

Describing the Relation Between Two Variables: Correlation, Regression, and Contingency Tables

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