Q: Using the 1.5*IQR criterion for outliers, what is the upper fence if $$Q_3 = 90$ and $$IQR = 18$$?

$$Upper\ Fence = Q_3$ + 1.5 \times IQR = 90 + 1.5 \times 18 = 117$$

Q: Using the regression equation $$Y = -0.47X + 24.3$$, what is the predicted vitamin A content for a carrot cooked for 25 minutes?

$$Y = -0.47 \times 25 + 24.3 = 12.55$$

Q: A random sample of 125 students is taken, and $$X is $$the number who completed an IB course in high school. If the probability of a student having completed an IB course is $$p = 0.2$$, what is the mean of $$X$$?

$$E[X] = np = 125 \times 0.2 = 25$$

Q: For the same scenario, what is the standard deviation of $$X$$?

$$\sigma = \sqrt{np(1-p)} = \sqrt{125 \times 0.2 \times 0.8} \approx 4.472$$

Question 1

Which of the following variables from the table is a qualitative (categorical) variable that takes numeric values?

Accepted Answer

Zip Code

Question 2

A histogram of tip amounts at a restaurant shows a longer tail to the right. Which of the following best describes the distribution and the relationship between the mean and median?

Accepted Answer

The distribution is skewed right, and the mean is greater than the median.

Question 3

Given $$Q_1 = 72$ and Q_3$ = 90$$, what is the interquartile range (IQR)?

Accepted Answer

$$IQR = Q_3$ - Q_1$ = 90 - 72 = 18$$

Question 4

Using the 1.5*IQR criterion for outliers, what is the upper fence if $$Q_3 = 90$ and $$IQR = 18$$?

Accepted Answer

$$Upper\ Fence = Q_3$ + 1.5 	imes IQR = 90 + 1.5 	imes 18 = 117$$

Question 5

A study finds the least-squares regression equation for vitamin A content in carrots as $$Y = -0.47X + 24.3. $$What does the slope of $$-0.47$$ mean in this context?

Accepted Answer

For each additional minute cooked, the vitamin A content decreases by 0.47 mg.

Question 6

Using the regression equation $$Y = -0.47X + 24.3$$, what is the predicted vitamin A content for a carrot cooked for 25 minutes?

Accepted Answer

$$Y = -0.47 	imes 25 + 24.3 = 12.55$$

Question 7

If a histogram of high school GPAs shows the highest bar at GPA 4.0, what can you infer about the distribution?

Accepted Answer

The distribution is left-skewed.

Question 8

Given the probability mass function $$P(X = x) = \frac{17 - x}{55}$$ for $$x = 2, 4, 6, 8, 10$$, what is $$P(X \u003e 7)$$?

Accepted Answer

$$P(X = 8) + P(X = 10) = \frac{9}{55} + \frac{7}{55} = \frac{16}{55} \approx 0.291$$

Question 9

For the same probability mass function $$P(X = x) = \frac{17 - x}{55}$$ for $$x = 2, 4, 6, 8, 10$$, what is the mean (expected value) of $$X$$?

Accepted Answer

$$E[X] = \sum x \cdot P(X = x) = 2 \cdot \frac{15}{55} + 4 \cdot \frac{13}{55} + 6 \cdot \frac{11}{55} + 8 \cdot \frac{9}{55} + 10 \cdot \frac{7}{55} = 5.6$$

Question 10

A random sample of 125 students is taken, and $$X is $$the number who completed an IB course in high school. If the probability of a student having completed an IB course is $$p = 0.2$$, what is the mean of $$X$$?

Accepted Answer

$$E[X] = np = 125 	imes 0.2 = 25$$

Question 11

For the same scenario, what is the standard deviation of $$X$$?

Accepted Answer

$$\sigma = \sqrt{np(1-p)} = \sqrt{125 	imes 0.2 	imes 0.8} \approx 4.472$$

Question 12

A fair six-sided die and a spinner with values 3, 4, and 5 are used. $$X is $$the sum of the die and spinner. What is the probability that $$X = 7$$?

Accepted Answer

$$P(X = 7) = \frac{6}{18} = \frac{1}{3}$$

Question 13

A scratch-off ticket has the following probabilities: $$P(\0) = 0.625$, P$$(\$$5) = 0.225$$, $$P(\10) = 0.0725$, P$$(\$$25) = 0.0625$$, $$P(\50) = 0.015$. What is the probability of winning at least 10$$?

Accepted Answer

$$P(\10) + P$$(\$$25) + P(\50) = 0.0725 + 0.0625 + 0.015 = 0.15$

Question 14

For the scratch-off ticket, what is the expected value (mean) of the prize money $$X$$?

Accepted Answer

$$E[X] = 0 	imes 0.625 + 5 	imes 0.225 + 10 	imes 0.0725 + 25 	imes 0.0625 + 50 	imes 0.015 = 3.25$$

Question 15

Shelby’s scatterplot shows a very weak negative linear relationship, but her $$r$$ value is $$0.758. $$What is the most likely explanation?

Accepted Answer

Shelby made a calculation error; a weak negative relationship should have an $$r$$ closer to 0 and negative.

Question 16

Kayla’s scatterplot shows a very strong negative linear relationship, but her $$r$$ value is $$-1.356. $$What is the issue with this $$r$$ value?

Accepted Answer

The value of $$r$$ must be between $$-1$$ and $$1$$; $$-1.356 is $$not possible.

Question 17

If a distribution is left-skewed, which of the following is most likely true?

Accepted Answer

The mean is less than the median.

STAT100 Exam 1 Study Guide: Key Concepts in Introductory Statistics

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