Q: Suppose the first quartile $$Q_1 is 2.2$$, the third quartile $$Q_3 is 9.4$$, and the interquartile range is 7.2. What is the upper fence for outlier detection?

$$Q_3 + 1.5$$ \times IQR = 9.4 + 1.5 \times 7.2 = 20.2$$

Question 1

Which of the following best describes the difference between the population mean $$\mu$$ and the sample mean $$\overline{x}$$?

Accepted Answer

The population mean $$\mu is a $$parameter calculated from all individuals in a population, while the sample mean $$\overline{x} is a $$statistic calculated from a subset (sample) of the population.

Question 2

Given the following data set: 5, 7, 8, 12, 15, what is the median?

Accepted Answer

$$8$$

Question 3

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

Accepted Answer

Median

Question 4

A data set has the following values: 2, 2, 3, 4, 5, 5, 5, 6. What is the mode?

Accepted Answer

$$5$$

Question 5

If a distribution is skewed right, which measure of central tendency is typically greater?

Accepted Answer

Mean

Question 6

Given the exam scores: 62, 68, 71, 74, 77, 82, 84, 88, 90, 94, what is the range?

Accepted Answer

$$94 - 62 = 32$$

Question 7

Which formula correctly represents the population standard deviation $$\sigma$$ for $$N$$ observations $$x_1$$, $$x_2$$, ..., x_N$$ with mean $$\mu$$?

Accepted Answer

$$\sigma = \sqrt{\frac{\sum_{i=1}^N (x_i - \mu)^2$}{N}}$$

Question 8

A sample of four students has scores: 62, 68, 77, 88. The sample mean is $$73.75. $$What is the sample variance $$s^2$$?

Accepted Answer

$$s^2$$ = \frac{$$(62-73.75)^2$$ + $$(68-73.75)^2$$ + $$(77-73.75)^2$$ + $$(88-73.75)^2$$}{4-1}$$

Question 9

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

Accepted Answer

$$95\%$$

Question 10

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

Accepted Answer

$$75\%$$

Question 11

Given a frequency distribution with midpoints $$m_i$$ and frequencies $$f_i$$, which formula approximates the sample mean $$\overline{x}$$?

Accepted Answer

$$\overline{x} = \frac{\sum_{i=1}^n m_i f_i}{\sum_{i=1}^n f_i}$$

Question 12

A student’s test score is 1.5 standard deviations above the mean. What is the student’s z-score?

Accepted Answer

$$z = 1.5$$

Question 13

If a value is at the 90th percentile in a data set, what does this mean?

Accepted Answer

90% of the data values are less than or equal to this value.

Question 14

Which formula is used to calculate the interquartile range (IQR)?

Accepted Answer

$$IQR = Q_3$ - Q_1$

Question 15

A boxplot shows a right whisker much longer than the left, and the median is left of center in the box. What does this indicate about the distribution?

Accepted Answer

The distribution is skewed right.

Question 16

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

Accepted Answer

The mean

Question 17

Suppose the first quartile $$Q_1 is 2.2$$, the third quartile $$Q_3 is 9.4$$, and the interquartile range is 7.2. What is the upper fence for outlier detection?

Accepted Answer

$$Q_3 + 1.5$$ 	imes IQR = 9.4 + 1.5 	imes 7.2 = 20.2$$

Question 18

In a side-by-side boxplot comparing ride distances for 'Shared' and 'Not Shared' rides, the 'Not Shared' group has a higher median and more dispersion. What does this suggest?

Accepted Answer

People tend to share rides for shorter distances, and not shared rides vary more in distance.

Question 19

Which of the following is the correct formula for a weighted mean $$\overline{x}_w$$ with values $$x_i$$ and weights $$w_i$$?

Accepted Answer

$$\overline{x}_w = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i}$$

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

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