Q: Given the following grouped data: Class midpoints $$x_i = 3, 8, 13$$, frequencies $$f_i = 10, 20, 30$$, what is the approximate mean?

$$\frac{3 \times 10 + 8 \times 20 + 13 \times 30}{10 + 20 + 30} = \frac{30 + 160 + 390}{60} = \frac{580}{60} \approx 9.67$$

Q: If the first quartile $$Q_1 is 28 $$and the third quartile $$Q_3 is 38$$, what are the lower and upper fences for outliers?

Lower fence: $$28 - 1.5 \times 10 = 13$$; Upper fence: $$38 + 1.5 \times 10 = 53$$

Q: A weighted mean is calculated for a mix of nuts: 1 lb at $$1.25, 2 lb at 3.25$$, and 1.5 lb at $5.40 per pound. What is the cost per pound?

$$\frac{1 \times 1.25 + 2 \times 3.25 + 1.5 \times 5.40}{1 + 2 + 1.5} = \frac{15.85}{4.5} = 3.52$$

Q: A data set has $$Q_1 = 20$, Q_3$ = 40$$, and an observation at 70. Is 70 an outlier?

No, because $$70 = 40 + 1.5 \times 20 = 70$$

Question 1

Suppose the travel times (in minutes) to work for seven employees are: 23, 36, 23, 18, 5, 26, 43. What is the population mean travel time?

Accepted Answer

$$\frac{23 + 36 + 23 + 18 + 5 + 26 + 43}{7} = 24.9$$

Question 2

Given the data set: 5, 18, 23, 23, 26, 36, 43, what is the median?

Accepted Answer

$$23$$

Question 3

Which measure of central tendency is most resistant to extreme values (outliers)?

Accepted Answer

Median

Question 4

A data set has a mean of 50 and a median of 60. Which best describes the shape of the distribution?

Accepted Answer

Skewed left

Question 5

What is the formula for the sample standard deviation $$s$$ for a data set $$x_1$$, $$x_2$$, ..., x_n$$ with sample mean $$\bar{x}$$?

Accepted Answer

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

Question 6

If a distribution is bell-shaped with mean $$\mu$$ and standard deviation $$\sigma$$, approximately what percentage of data falls within $$\mu \pm 2\sigma$$?

Accepted Answer

$$95\%$$

Question 7

Given the following grouped data: Class midpoints $$x_i = 3, 8, 13$$, frequencies $$f_i = 10, 20, 30$$, what is the approximate mean?

Accepted Answer

$$\frac{3 	imes 10 + 8 	imes 20 + 13 	imes 30}{10 + 20 + 30} = \frac{30 + 160 + 390}{60} = \frac{580}{60} \approx 9.67$$

Question 8

A student scores 85 on a test with mean 75 and standard deviation 5. What is the student's z-score?

Accepted Answer

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

Question 9

Which of the following is the correct formula for the interquartile range (IQR)?

Accepted Answer

$$IQR = Q_3$ - Q_1$

Question 10

If the first quartile $$Q_1 is 28 $$and the third quartile $$Q_3 is 38$$, what are the lower and upper fences for outliers?

Accepted Answer

Lower fence: $$28 - 1.5 	imes 10 = 13$$; Upper fence: $$38 + 1.5 	imes 10 = 53$$

Question 11

A boxplot shows the median closer to the lower quartile and a longer upper whisker. What is the likely shape of the distribution?

Accepted Answer

Skewed right

Question 12

Which measure of central tendency should be used for a distribution that is highly skewed right?

Accepted Answer

Median

Question 13

If a data set has no repeated values, what can be said about its mode?

Accepted Answer

There is no mode

Question 14

According to Chebyshev’s Inequality, at least what percentage of data lies within $$k=2$$ standard deviations of the mean for any distribution?

Accepted Answer

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

Question 15

A weighted mean is calculated for a mix of nuts: 1 lb at $$1.25, 2 lb at 3.25$$, and 1.5 lb at $5.40 per pound. What is the cost per pound?

Accepted Answer

$$\frac{1 	imes 1.25 + 2 	imes 3.25 + 1.5 	imes 5.40}{1 + 2 + 1.5} = \frac{15.85}{4.5} = 3.52$$

Question 16

Which of the following best describes the relationship between mean and median for a symmetric distribution?

Accepted Answer

Mean is roughly equal to median

Question 17

If the five-number summary of a data set is: 6.5, 12.0, 13.6, 14.4, 14.5, what is the interquartile range?

Accepted Answer

$$14.4 - 12.0 = 2.4$$

Question 18

Which of the following is NOT a step in constructing a boxplot?

Accepted Answer

Calculate the mean and plot it as the center of the box

Question 19

A data set has $$Q_1 = 20$, Q_3$ = 40$$, and an observation at 70. Is 70 an outlier?

Accepted Answer

No, because $$70 = 40 + 1.5 	imes 20 = 70$$

Numerically Summarizing Data: Measures of Central Tendency, Dispersion, and Position

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