Question 1

Which of the following statements best describes the range as a measure of variability in a data set?

Accepted Answer

The range is the difference between the largest and smallest measurements in a data set.

Question 2

Given the following sample: 2, 3, 3, 3, 4, what is the sample variance $$s^2$$? Use the formula $$s^2$$ = \frac{\sum_{i=1}^n (x_i - \bar{x}$$)^2$$}{n-1}$$.

Accepted Answer

$$s^2 = 0.5$

Question 3

A manufacturer claims the average life of its batteries is 60 months with a standard deviation of 10 months. According to the Empirical Rule, approximately what percentage of batteries will last between 50 and 70 months?

Accepted Answer

68%

Question 4

Which of the following is TRUE about the Empirical Rule for a mound-shaped, symmetric distribution?

Accepted Answer

Approximately 95% of the data falls within two standard deviations of the mean.

Question 5

If a student scores at the 90th percentile on an exam, which of the following is correct?

Accepted Answer

The student scored higher than 90% of the students who took the exam.

Question 6

Given a sample mean $$\bar{x} = 540$$ and sample standard deviation $$s = 100$$, what is the z-score for a student who scored 440?

Accepted Answer

$$z = -1$$

Question 7

Which of the following best describes the meaning of a z-score of -2.5 in a mound-shaped distribution?

Accepted Answer

The value is 2.5 standard deviations below the mean and is considered surprisingly low.

Question 8

Which of the following is NOT a property of the sample standard deviation $$s$$?

Accepted Answer

It is always negative.

Question 9

A company’s yearly sales are at the 90th percentile among oil companies. What does this indicate about the company’s sales compared to others?

Accepted Answer

The company’s sales are higher than 90% of other oil companies.

Question 10

Suppose the profit margin per job for two cost estimators is shown in the histograms above. Which estimator has more variability in profit margins?

Accepted Answer

Estimator A, because the data is more spread out across profit margins.

Question 11

If a data set has a mean of 20 and a standard deviation of 4, what interval contains approximately 95% of the data according to the Empirical Rule?

Accepted Answer

$$(12, 28)$$

Question 12

Which of the following is the correct formula for the sample variance $$s^2$$ using raw scores?

Accepted Answer

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

Question 13

A data set is highly skewed to the right. Which measure of central tendency is most likely to be greater than the others?

Accepted Answer

Mean

Question 14

If a company’s R&D percentage is at the 25th percentile among 50 companies, what does this mean?

Accepted Answer

The company’s R&D percentage is higher than 25% of the companies.

Question 15

Which of the following best describes why the range is considered an insensitive measure of variability for large data sets?

Accepted Answer

It only considers the two extreme values and ignores all other data.

Question 16

If a value in a data set has a z-score of 0, what does this indicate?

Accepted Answer

The value is equal to the mean.

Question 17

A student’s test score is two standard deviations above the mean. According to the Empirical Rule, what percentage of students scored lower than this student?

Accepted Answer

Approximately 97.5%

Numerical Measures of Variability and Relative Standing in Business Statistics

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