Q: Given the regression equation $$\text{mpg} = 40 - 0.005 \times \text{weight}$$, what is the predicted mpg for a car weighing 2,000 kg?

$$\text{mpg} = 40 - 0.005 \times 2000 = 30$$

Q: Suppose $$P(C) = 0.05 is $$the probability a student cheats, and $$P(A|C) = 1 is $$the probability a student gets the correct answer if they cheat. What is the probability a student both cheats and gets the correct answer?

$$P(C \cap A) = P(C) \times P(A|C) = 0.05 \times 1 = 0.05$$

Q: If two events $$A$$ and $$B$$ are independent, which of the following is true?

$$P(A \cap B) = P(A) \times P(B)$$

Question 1

A boxplot of the variable 'temperature' shows a long tail to the right and one point far from the rest of the data. Which of the following best describes the distribution?

Accepted Answer

The distribution is skewed to the right and contains an outlier.

Question 2

Suppose the mean car weight in a sample is 1,200 kg and the standard deviation is 100 kg. If a car weighs 1,400 kg, what is its z-score?

Accepted Answer

$$z = \frac{1400 - 1200}{100} = 2$$

Question 3

A scatterplot of car weight (x) and mpg (y) shows a negative linear relationship. What does this imply?

Accepted Answer

As car weight increases, mpg tends to decrease.

Question 4

If the correlation coefficient $$r$$ between two variables is $$-0.85$$, which of the following is true?

Accepted Answer

There is a strong negative linear relationship between the variables.

Question 5

Given the regression equation $$	ext{mpg} = 40 - 0.005 	imes 	ext{weight}$$, what is the predicted mpg for a car weighing 2,000 kg?

Accepted Answer

$$	ext{mpg} = 40 - 0.005 	imes 2000 = 30$$

Question 6

Which of the following statements about $$R^2 in $$linear regression is correct?

Accepted Answer

$$R^2$$ measures the proportion of variability in the response variable explained by the model.

Question 7

A student randomly guesses on a multiple-choice question with 4 options. What is the probability they choose the correct answer?

Accepted Answer

$$P = \frac{1}{4} = 0.25$$

Question 8

If $$P(A) = 0.3$$, $$P(B) = 0.4$$, and $$P(A \cap B) = 0.12$$, what is $$P(A \cup B)$$?

Accepted Answer

$$P(A \cup B) = 0.3 + 0.4 - 0.12 = 0.58$$

Question 9

Which of the following is a categorical (qualitative) variable?

Accepted Answer

Car color

Question 10

Suppose $$P(C) = 0.05 is $$the probability a student cheats, and $$P(A|C) = 1 is $$the probability a student gets the correct answer if they cheat. What is the probability a student both cheats and gets the correct answer?

Accepted Answer

$$P(C \cap A) = P(C) 	imes P(A|C) = 0.05 	imes 1 = 0.05$$

Question 11

If two events $$A$$ and $$B$$ are independent, which of the following is true?

Accepted Answer

$$P(A \cap B) = P(A) 	imes P(B)$$

Question 12

A regression model is built to predict mpg from car weight, but the data only includes cars weighing 1,000 to 6,000 lbs. Why is it not appropriate to use this model to predict mpg for a car weighing 10,000 lbs?

Accepted Answer

The model is only valid within the range of observed data; extrapolation can lead to unreliable predictions.

Question 13

Which formula correctly calculates the sample variance $$s^2$$ for a dataset $$x_1$$, $$x_2$$, ..., x_n$$?

Accepted Answer

$$s^2$$ = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x}$$)^2$$

Question 14

If $$P(A) = 0.2$$ and $$P(B|A) = 0.5$$, what is $$P(A \cap B)$$?

Accepted Answer

$$P(A \cap B) = P(A) 	imes P(B|A) = 0.2 	imes 0.5 = 0.1$$

Question 15

Which of the following best describes the difference between a parameter and a statistic?

Accepted Answer

A parameter describes a population, while a statistic describes a sample.

Question 16

A histogram of exam scores is symmetric and bell-shaped. Which measure of center and spread is most appropriate to summarize the data?

Accepted Answer

Mean and standard deviation

Question 17

If $$P(A) = 0.6$$, $$P(B) = 0.5$$, and $$A$$ and $$B$$ are mutually exclusive, what is $$P(A \cap B)$$?

Accepted Answer

$$P(A \cap B) = 0$$

Question 18

Which of the following is NOT a property of the correlation coefficient $$r$$?

Accepted Answer

$$r$$ can be used to describe nonlinear relationships.

Question 19

A researcher records the number of siblings each student has. What type of variable is this?

Accepted Answer

Quantitative discrete

Statistics Midterm 1: Key Concepts and Applications

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