Q: Given the regression equation $$\hat{y} = 58.9 - 0.00749x$$, where $$x$$ represents the weight of a car in pounds, what is the predicted highway fuel consumption (in miles per gallon) for a car that weighs 3000 lbs?

$$\hat{y} = 58.9 - 0.00749 \times 3000 = 36.43$$

Q: Given the regression equation $$\hat{y} = -212 + 61.9x$$, where $$x is $$the head width (in inches) of a bear, what is the best predicted weight for a bear with a head width of 6.5 inches?

$$\hat{y} = -212 + 61.9 \times 6.5 = 190.35$$

Q: Given the regression equation $$\hat{y} = 53.7 - 0.0056x$$ for predicting highway fuel consumption $$y$$ from car weight $$x$$, what is the best predicted value for a car that weighs 2500 lbs?

$$\hat{y} = 53.7 - 0.0056 \times 2500 = 39.7$$

Question 1

Suppose a study finds a strong linear correlation between the concentration of carbon dioxide (CO$$_2) in $$the atmosphere and the global mean temperature. Does this necessarily mean that changes in CO$$_2$$ cause changes in global mean temperature? Why or why not?

Accepted Answer

No, because a strong correlation does not imply causation; there may be other variables involved.

Question 2

Given the regression equation $$\hat{y} = 58.9 - 0.00749x$$, where $$x$$ represents the weight of a car in pounds, what is the predicted highway fuel consumption (in miles per gallon) for a car that weighs 3000 lbs?

Accepted Answer

$$\hat{y} = 58.9 - 0.00749 	imes 3000 = 36.43$$

Question 3

Which of the following is the correct formula for the Pearson correlation coefficient $$r$$?

Accepted Answer

$$r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{n(\sum x^2$) - (\sum x)^2$} \sqrt{n(\sum y^2$) - (\sum y)^2$}}$$

Question 4

A psychologist tells you that in an ESP experiment, there is a 20% chance of answering a question correctly. What is the probability of answering incorrectly?

Accepted Answer

$$0.80$$

Question 5

If a multiple-choice question has five possible answers and only one is correct, what is the probability of guessing the wrong answer?

Accepted Answer

$$0.80$$

Question 6

Given the regression equation $$\hat{y} = -212 + 61.9x$$, where $$x is $$the head width (in inches) of a bear, what is the best predicted weight for a bear with a head width of 6.5 inches?

Accepted Answer

$$\hat{y} = -212 + 61.9 	imes 6.5 = 190.35$$

Question 7

In a study, the correlation coefficient between bear weight and chest size is $$r = 0.963141$$ with a p-value of $$0.000. $$What does this indicate?

Accepted Answer

There is a strong, statistically significant linear correlation between bear weight and chest size.

Question 8

If the probability that a randomly selected subject developed the flu is $$0.0875$$, and the probability that a subject received the vaccine treatment and developed the flu is $$0.00875$$, what is the probability that a randomly selected subject who developed the flu received the vaccine treatment?

Accepted Answer

$$\frac{0.00875}{0.0875} = 0.1$$

Question 9

If two undergraduate students are randomly selected and the probability that each takes online courses only is $$0.10$$, what is the probability that both take online courses only?

Accepted Answer

$$0.10 	imes 0.10 = 0.01$$

Question 10

Given the regression equation $$\hat{y} = 53.7 - 0.0056x$$ for predicting highway fuel consumption $$y$$ from car weight $$x$$, what is the best predicted value for a car that weighs 2500 lbs?

Accepted Answer

$$\hat{y} = 53.7 - 0.0056 	imes 2500 = 39.7$$

Question 11

A table shows that the probability of a randomly selected person being female is $$0.4$$ and the probability of being left-handed is $$0.1. If $$these events are independent, what is the probability that a randomly selected person is both female and left-handed?

Accepted Answer

$$0.4 	imes 0.1 = 0.04$$

Question 12

If the correlation coefficient $$r$$ between two variables is close to zero, what does this indicate about the relationship between the variables?

Accepted Answer

There is little to no linear relationship between the variables.

Question 13

Suppose the probability that a randomly selected day is the author's birthday is $$\frac{1}{365}. $$What is the probability that it is not the author's birthday?

Accepted Answer

$$1 - \frac{1}{365} = \frac{364}{365}$$

Question 14

A study finds a strong linear correlation between the number of PhDs awarded in civil engineering and the per capita consumption of mozzarella cheese. What is the most reasonable interpretation?

Accepted Answer

The correlation is likely coincidental and does not imply causation.

Question 15

If the probability of an event $$A is$$ $$0.3$$, what is the probability of the complement of $$A (i.e$$., $$A'$$)?

Accepted Answer

$$1 - 0.3 = 0.7$$

Question 16

Given the regression equation $$\hat{y} = 53.7 - 0.0056x$$ and $$r = -0.95$$, what does the negative value of $$r$$ indicate about the relationship between car weight and highway fuel consumption?

Accepted Answer

As car weight increases, highway fuel consumption decreases.

Question 17

If the probability of a subject developing the flu after receiving a placebo is $$\frac{56}{493}$$, what is this probability as a decimal rounded to three decimal places?

Accepted Answer

$$0.114$$

Question 18

If the correlation coefficient between two variables is $$r = 0.7$$, what can be said about the strength and direction of the relationship?

Accepted Answer

There is a strong positive linear relationship.

Correlation, Causation, and Linear Regression in Statistics

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