Q: Given the following frequency distribution for the number of no-shows on flights: 0 (37 flights), 1 (31 flights), 2 (20 flights), 3 (16 flights), 4 (12 flights), 5 (4 flights), what is the weighted mean number of no-shows per flight?

$$\frac{(0 \times 37) + (1 \times 31) + (2 \times 20) + (3 \times 16) + (4 \times 12) + (5 \times 4)}{37 + 31 + 20 + 16 + 12 + 4} = 1.56$$

Q: A company wants to compare the consistency of two machines. Machine A has a mean output of 100 units and a standard deviation of 5 units. Machine B has a mean output of 200 units and a standard deviation of 12 units. Which machine is more consistent based on the coefficient of variation?

Machine A: $$CV = \frac{5}{100} \times 100 = 5\%$$, Machine B: $$CV = \frac{12}{200} \times 100 = 6\%. $$Machine A is more consistent.

Q: Chebyshev’s Theorem states that for any data set, at least what percentage of values will fall within 3 standard deviations of the mean?

At least $$\left(1 - \frac{1}{3^2$}\right) \times 100 = 88.89\%$$

Q: A student’s score is at the 90th percentile in a class of 20 students. How many students scored below this student?

$$0.9 \times 20 = 18$$ students

Q: A company’s customer satisfaction scores at four locations are 7.2 (116 customers), 8.8 (80 customers), 6.0 (67 customers), and 7.5 (91 customers). What is the weighted mean satisfaction score?

$$\frac{(116 \times 7.2) + (80 \times 8.8) + (67 \times 6.0) + (91 \times 7.5)}{116 + 80 + 67 + 91} = 7.41$$

Q: If the mean and standard deviation of a population are $$\mu = 100$$ and $$\sigma = 15$$, what value corresponds to a z-score of 2?

$$x = \mu + z\sigma = 100 + 2 \times 15 = 130$$

Question 1

A company has five machines producing widgets. The number of widgets produced by each machine in a day is: 250, 300, 275, 325, and 350. What is the sample mean number of widgets produced per machine?

Accepted Answer

$$\frac{250 + 300 + 275 + 325 + 350}{5} = 300$$

Question 2

A real estate agent is analyzing house prices: $$100,000, 100$$,000, $$300,000, 500$$,000, $$2,000,000. Which measure of central tendency is LEAST affected by the 2$$,000,000 outlier?

Accepted Answer

Median

Question 3

Given the following frequency distribution for the number of no-shows on flights: 0 (37 flights), 1 (31 flights), 2 (20 flights), 3 (16 flights), 4 (12 flights), 5 (4 flights), what is the weighted mean number of no-shows per flight?

Accepted Answer

$$\frac{(0 	imes 37) + (1 	imes 31) + (2 	imes 20) + (3 	imes 16) + (4 	imes 12) + (5 	imes 4)}{37 + 31 + 20 + 16 + 12 + 4} = 1.56$$

Question 4

Which of the following is the formula for the sample variance $$s^2$$?

Accepted Answer

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

Question 5

A sample of exam scores is: 70, 75, 80, 85, 90. What is the range of the scores?

Accepted Answer

$$90 - 70 = 20$$

Question 6

A company wants to compare the consistency of two machines. Machine A has a mean output of 100 units and a standard deviation of 5 units. Machine B has a mean output of 200 units and a standard deviation of 12 units. Which machine is more consistent based on the coefficient of variation?

Accepted Answer

Machine A: $$CV = \frac{5}{100} 	imes 100 = 5\%$$, Machine B: $$CV = \frac{12}{200} 	imes 100 = 6\%. $$Machine A is more consistent.

Question 7

If a data set is left-skewed, which of the following is true about the relationship between the mean and the median?

Accepted Answer

Mean \u003c Median

Question 8

A student scored 85 on a test where the mean was 80 and the standard deviation was 5. What is the student's z-score?

Accepted Answer

$$z = \frac{85 - 80}{5} = 1$$

Question 9

According to the Empirical Rule, approximately what percentage of data falls within two standard deviations of the mean in a bell-shaped distribution?

Accepted Answer

95%

Question 10

Chebyshev’s Theorem states that for any data set, at least what percentage of values will fall within 3 standard deviations of the mean?

Accepted Answer

At least $$\left(1 - \frac{1}{3^2$}ight) 	imes 100 = 88.89\%$$

Question 11

A data set has Q1 = 20, Q3 = 40. What is the interquartile range (IQR)?

Accepted Answer

$$IQR = Q3 - Q1 = 40 - 20 = 20$$

Question 12

Given a data set: 2, 4, 4, 4, 5, 5, 7, 9. What is the mode?

Accepted Answer

4

Question 13

A boxplot shows the median closer to the lower quartile (Q1) than the upper quartile (Q3). What does this indicate about the distribution?

Accepted Answer

The distribution is right-skewed (positively skewed)

Question 14

A student’s score is at the 90th percentile in a class of 20 students. How many students scored below this student?

Accepted Answer

$$0.9 	imes 20 = 18$$ students

Question 15

Which Excel function would you use to calculate the sample standard deviation for a data set in column A from row 2 to row 10?

Accepted Answer

=STDEV.S(A2:A10)

Question 16

A company’s customer satisfaction scores at four locations are 7.2 (116 customers), 8.8 (80 customers), 6.0 (67 customers), and 7.5 (91 customers). What is the weighted mean satisfaction score?

Accepted Answer

$$\frac{(116 	imes 7.2) + (80 	imes 8.8) + (67 	imes 6.0) + (91 	imes 7.5)}{116 + 80 + 67 + 91} = 7.41$$

Question 17

Which of the following is NOT a measure of central tendency?

Accepted Answer

Range

Question 18

If the mean and standard deviation of a population are $$\mu = 100$$ and $$\sigma = 15$$, what value corresponds to a z-score of 2?

Accepted Answer

$$x = \mu + z\sigma = 100 + 2 	imes 15 = 130$$

Descriptive Statistics: Measures of Central Tendency, Variation, and Relative Position

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