Q: A company wants to compare the consistency of two machines producing bolts. Machine A has a mean length of 10 cm and a standard deviation of 0.5 cm. Machine B has a mean length of 20 cm and a standard deviation of 1.2 cm. Which machine is more consistent based on the coefficient of variation?

Machine A, because $$CV = \frac{0.5}{10} \times 100 = 5\%$$ and Machine B, $$CV = \frac{1.2}{20} \times 100 = 6\%.$$

Question 1

A company wants to summarize the salaries of its employees. The data set contains a few extremely high salaries. Which measure of central tendency is most appropriate to use, and why?

Accepted Answer

The median, because it is not affected by outliers.

Question 2

Given the following sample data: $$12, 15$$, $$20, 22$$, $31. Calculate the sample mean using the correct formula.

Accepted Answer

$$\displaystyle \bar{x} = \frac{12 + 15 + 20 + 22 + 31}{5} = 20$$

Question 3

A student’s final grade is based on three components: Exam (score: 90, weight: 50%), Project (score: 85, weight: 30%), Homework (score: 100, weight: 20%). What is the weighted mean of the student’s final grade?

Accepted Answer

$$\displaystyle \bar{x} = \frac{(0.5)(90) + (0.3)(85) + (0.2)(100)}{0.5 + 0.3 + 0.2} = 91.5$$

Question 4

Which of the following statements about the mean is TRUE?

Accepted Answer

The mean is sensitive to outliers.

Question 5

A data set has the following values: 2, 4, 4, 7, 9, 10. What is the mode?

Accepted Answer

The mode is 4.

Question 6

Which formula correctly represents the sample variance?

Accepted Answer

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

Question 7

A sample of five donations to a charity are: $$10, 20$$, $$30, 40$$, $100. What is the range of the donations?

Accepted Answer

$90

Question 8

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

Accepted Answer

Mean \u003c Median

Question 9

A company wants to compare the consistency of two machines producing bolts. Machine A has a mean length of 10 cm and a standard deviation of 0.5 cm. Machine B has a mean length of 20 cm and a standard deviation of 1.2 cm. Which machine is more consistent based on the coefficient of variation?

Accepted Answer

Machine A, because $$CV = \frac{0.5}{10} 	imes 100 = 5\%$$ and Machine B, $$CV = \frac{1.2}{20} 	imes 100 = 6\%.$$

Question 10

A value in a data set has a z-score of -2.5. What does this indicate?

Accepted Answer

The value is 2.5 standard deviations below the mean.

Question 11

According to the empirical rule, what percentage of data falls within two standard deviations of the mean in a normal distribution?

Accepted Answer

Approximately 95%

Question 12

Which formula is used to estimate the mean from grouped data?

Accepted Answer

$$\displaystyle \bar{x}_g = \frac{\sum_{i=1}^k f_i m_i}{n}$$

Question 13

A student scored 85 on an exam where the mean was 75 and the standard deviation was 5. What is the student’s z-score?

Accepted Answer

$$z = \frac{85 - 75}{5} = 2$$

Question 14

Which of the following best describes the interquartile range (IQR)?

Accepted Answer

The difference between the third quartile and the first quartile.

Question 15

If a value is greater than $$Q3 + 1.5 	imes IQR$$, what is it considered?

Accepted Answer

An outlier

Question 16

Which formula is used to calculate the sample correlation coefficient between two variables x and y?

Accepted Answer

$$\displaystyle r_{xy} = \frac{s_{xy}}{s_x s_y}$$

Question 17

A box-and-whisker plot is constructed for the number of visitors to national parks. If the maximum value is an outlier, how is it represented on the plot?

Accepted Answer

With an asterisk (*) beyond the whisker

Question 18

Which of the following is NOT included in the five-number summary?

Accepted Answer

The mean

Question 19

If the sample correlation coefficient $$r_{xy}$$ between hours studied and exam grade is 0.95, what does this indicate?

Accepted Answer

There is a strong positive linear relationship between hours studied and exam grade.

Chapter 3: Calculating Descriptive Statistics – Business Statistics Study Notes

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