Q: Suppose a simple linear regression equation is $$ \hat{y} = 3.6 + 0.97x . $$What is the predicted value of $$ y $$ when $$ x = 40 $$?

$$ \hat{y} = 3.6 + 0.97 \times 40 = 42.4 $$

Question 1

Which of the following best describes the difference between an explanatory variable and a response variable in a statistical study?

Accepted Answer

The explanatory variable is the predictor, and the response variable is the outcome.

Question 2

A researcher finds that taller roller coasters tend to have higher maximum speeds. Which of the following statements is correct?

Accepted Answer

There is a correlation between roller coaster height and speed, but this does not necessarily mean height causes higher speed.

Question 3

Given the following contingency table for food type and pesticide residue presence, what is the conditional proportion of organic food with pesticide residues?

Organic: 29 present, 98 not present, total 127

Conventional: 19,485 present, 7,086 not present, total 26,571

Accepted Answer

$$ \frac{29}{127} = 0.228 $$

Question 4

Which graphical method is most appropriate for visualizing the relationship between two quantitative variables?

Accepted Answer

Scatterplot

Question 5

If the correlation coefficient $$ r $$ between two variables is $$ -1 $$, what does this indicate?

Accepted Answer

There is a perfect negative linear relationship.

Question 6

Suppose a simple linear regression equation is $$ \hat{y} = 3.6 + 0.97x . $$What is the predicted value of $$ y $$ when $$ x = 40 $$?

Accepted Answer

$$ \hat{y} = 3.6 + 0.97 	imes 40 = 42.4 $$

Question 7

In the regression equation $$ \hat{y} = 61.4 + 2.4x $$, what does the slope represent?

Accepted Answer

For each 1 unit increase in $$ x $$, $$ \hat{y} $$ increases by 2.4 units.

Question 8

A study finds a correlation of $$ r = 0.59 $$ between height and weight. What percent of the variation in weight is explained by height?

Accepted Answer

$$ r^2$ = 0.3481 $$, so 34.81\% of the variation is explained.

Question 9

Which of the following is NOT an assumption of simple linear regression?

Accepted Answer

The explanatory variable must be categorical.

Question 10

What is the main risk of extrapolation when using a regression equation?

Accepted Answer

Predictions may be unreliable because the relationship outside the observed data range is unknown.

Question 11

A scatterplot of ice cream sales and number of drownings shows a strong positive correlation. Which of the following best explains this relationship?

Accepted Answer

A lurking variable, such as season, affects both ice cream sales and drownings.

Question 12

Which of the following statements about influential outliers is TRUE?

Accepted Answer

Influential outliers can significantly change the slope of the regression line.

Question 13

Given the quadratic regression equation $$ \hat{y} = 13.62 + 54.05x - 4.719x^2$ $$, what is the predicted value of $$ y $$ when $$ x = 2.5 $$?

Accepted Answer

$$ \hat{y} = 13.62 + 54.05 	imes 2.5 - 4.719 	imes (2.5)^2$ = 13.62 + 135.125 - 29.49375 = 119.25125 $$

Question 14

Which of the following best describes a confounding variable?

Accepted Answer

A variable that is considered in a study and influences the relationship between $$ x $$ and $$ y .$$

Question 15

If the conditional proportions in a contingency table for two categorical variables are approximately the same across groups, what can you conclude?

Accepted Answer

There is no association between the two variables.

Question 16

Which of the following is the correct formula for calculating the risk of an outcome in a group?

Accepted Answer

$$ 	ext{Risk} = \frac{	ext{Number with trait}}{	ext{Total in group}} $$

Question 17

Which of the following is a correct interpretation of $$ R^2$ = 0.59  in a $$regression analysis of height and weight?

Accepted Answer

59\% of the variation in weight is explained by the linear relationship with height.

Question 18

Which of the following is NOT a reason why correlation does not imply causation?

Accepted Answer

Correlation always means one variable causes the other.

Question 19

A regression analysis is performed on a dataset with a clear curve in the scatterplot. Which model is likely more appropriate?

Accepted Answer

Quadratic regression

Exploring Relationships Between Variables: Chapter 3 Study Notes

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