Q: Chebyshev’s Inequality states that for any data set, at least what percentage of observations lie within $$k$$ standard deviations of the mean ($$k \u003e 1$$)?

$$\left(1 - \frac{1}{k^2$}\right) \times 100\%$$

Q: Which formula is used to check for outliers using quartiles and the interquartile range?

Lower fence: $$Q_1 - 1.5$$ \times \text{IQR}$$; Upper fence: Q_3$ + 1.5 \times \text{IQR}$$

Q: In a frequency distribution, how do you approximate the mean from grouped data?

$$\text{Mean} = \frac{\sum x_i f_i}{\sum f_i}$$, where $$x_i is $$the midpoint of each class and $$f_i is $$the frequency.

Question 1

Which of the following statements best describes a resistant statistic?

Accepted Answer

A statistic whose value does not change much when extreme values are present in the data set.

Question 2

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

Accepted Answer

23

Question 3

If a distribution is skewed right, which of the following is true about the mean and median?

Accepted Answer

Mean is substantially larger than the median.

Question 4

Which measure of central tendency is most appropriate to use when the data are qualitative?

Accepted Answer

Mode

Question 5

Given the following data: 23, 36, 23, 18, 5, 26, 43, what is the range?

Accepted Answer

$$43 - 5 = 38$$

Question 6

Which formula correctly calculates the population standard deviation for a set of $$N$$ values $$x_1$$, $$x_2$$, ..., x_N$$ with mean $$\mu$$?

Accepted Answer

$$\sigma = \sqrt{\frac{\sum (x_i - \mu)^2$}{N}}$$

Question 7

A sample of three travel times is: 5, 26, 36. The sample mean is $$22.333. $$What is the sample standard deviation?

Accepted Answer

$$s = \sqrt{\frac{300.432889 + 13.446889 + 186.786889}{2}} \approx 15.82$$

Question 8

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

Accepted Answer

95%

Question 9

If the mean waist circumference of 2-year-old males is $$48.5 cm $$and the standard deviation is $$4.8 cm$$, what interval contains approximately 99.7% of waist circumferences?

Accepted Answer

$$48.5 \pm 14.4$$

Question 10

Chebyshev’s Inequality states that for any data set, at least what percentage of observations lie within $$k$$ standard deviations of the mean ($$k \u003e 1$$)?

Accepted Answer

$$\left(1 - \frac{1}{k^2$}ight) 	imes 100\%$$

Question 11

What is the formula for a z-score for a sample value $$x$$ with sample mean $$\bar{x}$$ and sample standard deviation $$s$$?

Accepted Answer

$$z = \frac{x - \bar{x}}{s}$$

Question 12

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

Accepted Answer

The difference between the third and first quartiles.

Question 13

A data set has $$Q_1 = 18$, Q_3$ = 36. $$What is the interquartile range?

Accepted Answer

$$36 - 18 = 18$$

Question 14

Which of the following is NOT part of the five-number summary?

Accepted Answer

Mean

Question 15

If a data set is skewed left, which measure of central tendency is most appropriate to report?

Accepted Answer

Median

Question 16

Which formula is used to check for outliers using quartiles and the interquartile range?

Accepted Answer

Lower fence: $$Q_1 - 1.5$$ 	imes 	ext{IQR}$$; Upper fence: Q_3$ + 1.5 	imes 	ext{IQR}$$

Question 17

A boxplot is most useful for which of the following purposes?

Accepted Answer

Displaying the five-number summary and identifying outliers.

Question 18

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

Accepted Answer

Mean is roughly equal to median.

Question 19

In a frequency distribution, how do you approximate the mean from grouped data?

Accepted Answer

$$	ext{Mean} = \frac{\sum x_i f_i}{\sum f_i}$$, where $$x_i is $$the midpoint of each class and $$f_i is $$the frequency.

Chapter 3: Numerically Summarizing Data – Study Guide

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