Q: Chebyshev’s Inequality states that for any data set, at least what percentage of observations lie within $$k$$ standard deviations of the mean, where $$k \u003e 1$$? Use $$1 - \frac{1}{k^2$}.$$

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

Question 1

A researcher collects the following exam scores from a class of 10 students: 82, 77, 90, 71, 62, 68, 74, 84, 94, 88. What is the population mean of these scores? Use the formula $$\mu = \frac{\sum x_i}{N}.$$

Accepted Answer

$$\mu = 79$$

Question 2

Given the data set: 201, 257, 284, 208, 179, 222, 217, 206, 240 (song lengths in seconds), what is the median length? Use the median formula for an odd number of observations: $$	ext{Median} = 	ext{value at position } \frac{n+1}{2}$$ after sorting.

Accepted Answer

222 seconds

Question 3

Which of the following statements best describes a resistant statistic?

Accepted Answer

A statistic that is not substantially affected by extreme values in the data set.

Question 4

Yolanda's cell phone call lengths (in minutes) are: 1, 7, 4, 1, 2, 4, 3, 48, 3, 5, 3, 6. The mean is 7.25 and the median is 3.5. Which measure of central tendency better describes her typical call length, and why?

Accepted Answer

The median, because the mean is affected by the outlier (48 minutes).

Question 5

A data set has the following values: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 3. What is the mode?

Accepted Answer

0

Question 6

The IQ scores of two universities both have a mean of 100.0. University A has a standard deviation of 16.1, and University B has a standard deviation of 8.4. What does this indicate about the IQ distributions?

Accepted Answer

University A's IQ scores are more dispersed than University B's.

Question 7

Given the exam scores: 82, 77, 90, 71, 62, 68, 74, 84, 94, 88, what is the range? Use $$R = 	ext{max} - 	ext{min}.$$

Accepted Answer

$$R = 32$$

Question 8

Which formula correctly represents the population standard deviation $$\sigma$$ 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 9

A sample of 4 students has scores: 62, 88, 77, 68. What is the sample standard deviation $$s$$? Use $$s = \sqrt{\frac{\sum (x_i - \bar{x})^2$}{n-1}}.$$

Accepted Answer

$$s = 11.3$$

Question 10

If a data set is bell-shaped with mean $$\mu = 100$$ and standard deviation $$\sigma = 16.1$$, what percentage of data falls between $$\mu - 2\sigma$$ and $$\mu + 2\sigma$$ according to the Empirical Rule?

Accepted Answer

Approximately 95%

Question 11

Chebyshev’s Inequality states that for any data set, at least what percentage of observations lie within $$k$$ standard deviations of the mean, where $$k \u003e 1$$? Use $$1 - \frac{1}{k^2$}.$$

Accepted Answer

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

Question 12

A student’s SAT Math score is in the 74th percentile. What does this mean?

Accepted Answer

74% of students scored less than or equal to this score.

Question 13

Which of the following is the correct formula for a z-score for a population?

Accepted Answer

$$z = \frac{x - \mu}{\sigma}$$

Question 14

Given quartiles $$Q_1 = 735$ and Q_3$ = 4668$$ for a set of insurance claims, what is the interquartile range (IQR)? Use $$IQR = Q_3$ - Q_1$.

Accepted Answer

$$3933$$

Question 15

For the same insurance claim data, what is the upper fence for detecting outliers? Use $$	ext{Upper Fence} = Q_3$ + 1.5 	imes IQR$$ with $$Q_3 = 4668$ and $$IQR = 3933$$.

Accepted Answer

$$10,567.5$$

Question 16

Which of the following best describes the five-number summary?

Accepted Answer

Minimum, $$Q_1$$, Median, $$Q_3$$, Maximum

Question 17

A boxplot shows a long whisker on the right and the median closer to the left of the box. What does this indicate about the distribution?

Accepted Answer

The distribution is skewed right.

Question 18

Given grouped data with class midpoints $$x_i$$ and frequencies $$f_i$$, what is the formula to approximate the mean?

Accepted Answer

$$\bar{x} = \frac{\sum x_i f_i}{\sum f_i}$$

Question 19

A student’s z-score for a test is 2.0. What does this indicate about the student’s performance?

Accepted Answer

The student scored 2 standard deviations above the mean.

Chapter 3: Numerically Summarizing Data – Study Guide

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