Q: Given the following regression equation for car data: $$\hat{y} = 58.9 - 0.00749x$$, where $$x$$ represents weight, what is the predicted highway fuel consumption for a car that weighs 3000 lb?

Predicted value: $$\hat{y} = 58.9 - 0.00749 \times 3000 = 36.43 mi/$$gal

Q: If two undergraduate students are randomly selected and 10% of students take online courses only, what is the probability that both take online courses only?

Probability = $$0.10 \times 0.10 = 0.01$$

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

Predicted weight: $$\hat{y} = -212 + 61.9 \times 6.5 = 190.35 lb$$

Question 1

If a study finds a linear correlation between the concentration of carbon dioxide (CO$$_2) in $$the atmosphere and global mean temperature, does this prove that changes in CO$$_2$$ cause changes in global mean temperature? Why or why not?

Accepted Answer

No, correlation does not imply causation. There may be other factors influencing both variables.

Question 2

Given the following regression equation for car data: $$\hat{y} = 58.9 - 0.00749x$$, where $$x$$ represents weight, what is the predicted highway fuel consumption for a car that weighs 3000 lb?

Accepted Answer

Predicted value: $$\hat{y} = 58.9 - 0.00749 	imes 3000 = 36.43 mi/$$gal

Question 3

Which of the following best describes the meaning of the slope in the regression equation $$\hat{y} = 58.9 - 0.00749x$$?

Accepted Answer

For each additional pound of car weight, the predicted highway fuel consumption decreases by 0.00749 mi/gal.

Question 4

Using the Pearson correlation formula $$r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{n(\Sigma x^2$) - (\Sigma x)^2$} \sqrt{n(\Sigma y^2$) - (\Sigma y)^2$}}$$, what does a value of $$r = 0.963$$ indicate about the relationship between two variables?

Accepted Answer

There is a strong positive linear relationship between the two variables.

Question 5

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

Probability = $$1 - 0.20 = 0.80$$

Question 6

If a standard test has 5 possible answers for each question and a student guesses randomly, what is the probability of guessing incorrectly?

Accepted Answer

Probability = $$1 - \frac{1}{5} = 0.8$$

Question 7

Based on the vaccine experiment results (14 developed flu with vaccine, 1056 did not; 56 developed flu with placebo, 437 did not), what is the probability that a randomly selected subject developed flu?

Accepted Answer

Probability = $$\frac{14 + 56}{1602} = 0.0438$$

Question 8

If two undergraduate students are randomly selected and 10% of students take online courses only, what is the probability that both take online courses only?

Accepted Answer

Probability = $$0.10 	imes 0.10 = 0.01$$

Question 9

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

Accepted Answer

Predicted weight: $$\hat{y} = -212 + 61.9 	imes 6.5 = 190.35 lb$$

Question 10

If a randomly selected subject from the left-handed writing study is female, what is the probability she writes with her left hand?

Accepted Answer

Probability = $$\frac{23}{88} = 0.261$$

Question 11

In the cheese and engineering example, if a linear correlation is found between cheese consumption and civil engineering PhDs, does this mean eating more cheese causes people to earn PhDs in civil engineering?

Accepted Answer

No, correlation does not imply causation. The relationship may be coincidental or due to other factors.

Question 12

If the probability of randomly selecting a subject who developed flu and was given the vaccine treatment is $$\frac{14}{1602}$$, what is this value as a decimal?

Accepted Answer

Probability = $$0.0087$$

Question 13

Given the correlation coefficient $$r = 0.963$$ and a critical value $$r_{crit} = 0.2608 at$$ $$\alpha = 0.05$$, what can you conclude about the relationship between bear weight and chest size?

Accepted Answer

There is sufficient evidence to conclude a significant linear correlation between bear weight and chest size.

Question 14

If a randomly selected day of the year (not a leap year) is chosen, what is the probability it is the author's birthday?

Accepted Answer

Probability = $$\frac{1}{365} = 0.00274$$

Question 15

In a probability experiment, if $$P(A) = 0.3$$ and $$P(B) = 0.4$$, what is the probability that both events occur if they are independent?

Accepted Answer

Probability = $$0.3 	imes 0.4 = 0.12$$

Question 16

If the regression equation for small cars is $$\hat{y} = 53.7 - 0.00505x$$, what is the best predicted highway fuel consumption for a car that weighs 2500 lb?

Accepted Answer

Predicted value: $$\hat{y} = 53.7 - 0.00505 	imes 2500 = 41.075 mi/$$gal

Question 17

If the probability of randomly selecting a left-handed female is $$\frac{23}{200}$$, what is the probability of randomly selecting a left-handed male given there are 65 males and 15 left-handed males?

Accepted Answer

Probability = $$\frac{15}{200} = 0.075$$

Correlation, Causation, and Linear Regression in Statistics

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