Q: A company wants to compare the variation in salaries between two departments. Department A has a mean salary of $$50,000 and a standard deviation of 2$$,000. Department B has a mean salary of $$80,000 and a standard deviation of 4$$,000. Which department has greater relative variation?

Department B, because $$\frac{4000}{80000} \times 100\% = 5\%$$ and Department A is $$\frac{2000}{50000} \times 100\% = 4\%$$

Q: A teacher wants to identify outliers in the following data set of quiz scores: 5, 7, 8, 9, 10, 12, 15, 30. The first quartile $$Q_1 is 7$$, the third quartile $$Q_3 is 12. $$What is the upper boundary for outliers?

$$Q_3 + 1.5$$ \times ($$Q_3$$ - $$Q_1) = 12 + 1.5$$ \times 5 = 19.5$$

Q: A frequency distribution has the following class midpoints and frequencies: Midpoints: 10, 20, 30; Frequencies: 4, 6, 10. What is the estimated mean?

$$\bar{x} = \frac{(10 \times 4) + (20 \times 6) + (30 \times 10)}{4+6+10} = \frac{40+120+300}{20} = \frac{460}{20} = 23$$

Q: A student received grades of A (4 points) in 3 courses, B (3 points) in 2 courses, and C (2 points) in 1 course. What is the weighted mean (GPA)?

$$\frac{(4 \times 3) + (3 \times 2) + (2 \times 1)}{3+2+1} = \frac{12+6+2}{6} = \frac{20}{6} \approx 3.33$$

Question 1

A hospital administrator wants to summarize the ages of patients admitted in a week. She decides to use a frequency distribution with 6 classes. The minimum age is 18 and the maximum is 78. What is the appropriate class width to use?

Accepted Answer

$$\frac{78-18}{6} = 10$$; round up to 11, so class width is 11

Question 2

Which of the following best describes the difference between a frequency histogram and a frequency polygon?

Accepted Answer

A frequency histogram uses bars to represent frequencies, while a frequency polygon uses points connected by lines.

Question 3

A researcher collects the following data set: 5, 7, 7, 8, 10, 12, 12, 12, 15. What is the mode of this data set?

Accepted Answer

12

Question 4

A sample of 8 students has the following test scores: 70, 75, 80, 85, 90, 95, 100, 105. What is the median score?

Accepted Answer

87.5

Question 5

A company wants to compare the variation in salaries between two departments. Department A has a mean salary of $$50,000 and a standard deviation of 2$$,000. Department B has a mean salary of $$80,000 and a standard deviation of 4$$,000. Which department has greater relative variation?

Accepted Answer

Department B, because $$\frac{4000}{80000} 	imes 100\% = 5\%$$ and Department A is $$\frac{2000}{50000} 	imes 100\% = 4\%$$

Question 6

Given a data set with a mean of 100 and a standard deviation of 15, what is the z-score for a value of 130?

Accepted Answer

$$z = \frac{130-100}{15} = 2$$

Question 7

A teacher wants to identify outliers in the following data set of quiz scores: 5, 7, 8, 9, 10, 12, 15, 30. The first quartile $$Q_1 is 7$$, the third quartile $$Q_3 is 12. $$What is the upper boundary for outliers?

Accepted Answer

$$Q_3 + 1.5$$ 	imes ($$Q_3$$ - $$Q_1) = 12 + 1.5$$ 	imes 5 = 19.5$$

Question 8

A sample of 5 weights (in kg) is: 60, 62, 65, 68, 70. What is the sample mean?

Accepted Answer

$$\bar{x} = \frac{60+62+65+68+70}{5} = 65$$

Question 9

A frequency distribution has the following class midpoints and frequencies: Midpoints: 10, 20, 30; Frequencies: 4, 6, 10. What is the estimated mean?

Accepted Answer

$$\bar{x} = \frac{(10 	imes 4) + (20 	imes 6) + (30 	imes 10)}{4+6+10} = \frac{40+120+300}{20} = \frac{460}{20} = 23$$

Question 10

A population has the following data: 2, 4, 6, 8. What is the population variance?

Accepted Answer

$$\$$sigma^2$$ = \frac{(2-5)^2$ + (4-5)^2$ + (6-5)^2$ + (8-5)^2$}{4} = \frac{9+1+1+9}{4} = \frac{20}{4} = 5$$

Question 11

A sample of 10 exam scores has a mean of 80 and a standard deviation of 5. According to the Empirical Rule, approximately how many scores are expected to fall between 75 and 85?

Accepted Answer

About 68% of 10, so approximately 7 scores

Question 12

A pie chart is most appropriate for which type of data?

Accepted Answer

Qualitative data, showing proportions of categories

Question 13

A student received grades of A (4 points) in 3 courses, B (3 points) in 2 courses, and C (2 points) in 1 course. What is the weighted mean (GPA)?

Accepted Answer

$$\frac{(4 	imes 3) + (3 	imes 2) + (2 	imes 1)}{3+2+1} = \frac{12+6+2}{6} = \frac{20}{6} \approx 3.33$$

Question 14

Which of the following statements about the mean and median in a skewed right (positively skewed) distribution is correct?

Accepted Answer

The mean is greater than the median

Question 15

A dot plot is constructed for the number of books read by students in a month. Most dots are clustered between 2 and 5, with a single dot at 15. What does the dot at 15 represent?

Accepted Answer

An outlier

Question 16

A time series chart is best used to display which type of data?

Accepted Answer

Quantitative data measured at regular intervals over time

Question 17

A population has a mean of 50 and a standard deviation of 5. According to Chebychev’s Theorem, at least what percent of the data lies within 2 standard deviations of the mean?

Accepted Answer

At least $$1 - \frac{1}{2^2$} = 1 - \frac{1}{4} = 0.75 or 75$$%

Question 18

A box-and-whisker plot is constructed for the following data: Min=10, $$Q_1=15$$, Median=20, $$Q_3=25$$, Max=30. What percent of the data falls between $$Q_1$$ and $$Q_3$$?

Accepted Answer

About 50%

Descriptive Statistics: Frequency Distributions and Their Graphs

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