Q: For a population with $$\mu = 100$$ and $$\sigma = 15$$, what is the interval that contains approximately 99.7% of the data according to the Empirical Rule?

$$100 - 3 \times 15 to$$ $$100 + 3 \times 15 or$$ $$55 to$$ $$145$$

Q: A statistics exam has a mean score of 80 and a standard deviation of 15. Using Chebyshev’s Inequality, what is the interval that contains at least 60% of the scores?

$$(80 - 1.58 \times 15, 80 + 1.58 \times 15) = (56.3, 103.7)$$

Question 1

Which of the following statements best describes the range as a measure of dispersion?

Accepted Answer

It is the difference between the largest and smallest observations and is sensitive to outliers.

Question 2

Given the data set: 59, 71, 92, what is the range?

Accepted Answer

$$92 - 59 = 33$$

Question 3

Why is the range considered a limited measure of variation?

Accepted Answer

It ignores the distribution of the data and is based only on two observations.

Question 4

Which formula correctly represents the population variance?

Accepted Answer

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

Question 5

A sample of 5 students has scores: 70, 75, 80, 85, 90. What is the sample mean?

Accepted Answer

$$\bar{x} = \frac{70 + 75 + 80 + 85 + 90}{5} = 80$$

Question 6

Which formula is used to calculate the sample standard deviation?

Accepted Answer

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

Question 7

Why do we divide by $$n-1$$ instead of $$n$$ when calculating the sample variance?

Accepted Answer

To obtain an unbiased estimate of the population variance.

Question 8

If a distribution is bell-shaped, what percentage of data falls within 2 standard deviations of the mean according to the Empirical Rule?

Accepted Answer

Approximately 95%

Question 9

A hospital records the pulse rates of two patients three times a day. Patient A: 72, 74, 76. Patient B: 59, 71, 92. Both have the same mean. Which patient has greater variability and why?

Accepted Answer

Patient B, because the range is much larger.

Question 10

For a population with $$\mu = 100$$ and $$\sigma = 15$$, what is the interval that contains approximately 99.7% of the data according to the Empirical Rule?

Accepted Answer

$$100 - 3 	imes 15 to$$ $$100 + 3 	imes 15 or$$ $$55 to$$ $$145$$

Question 11

Using Chebyshev’s Inequality, what is the minimum percentage of data that must lie within 2 standard deviations of the mean for any distribution?

Accepted Answer

75%

Question 12

A statistics exam has a mean score of 80 and a standard deviation of 15. Using Chebyshev’s Inequality, what is the interval that contains at least 60% of the scores?

Accepted Answer

$$(80 - 1.58 	imes 15, 80 + 1.58 	imes 15) = (56.3, 103.7)$$

Question 13

Which statement about standard deviation is correct?

Accepted Answer

It represents the typical deviation from the mean.

Question 14

For a bell-shaped distribution, what percentage of data falls outside 2 standard deviations from the mean?

Accepted Answer

5%

Question 15

If a sample statistic consistently underestimates a population parameter, it is said to be:

Accepted Answer

Biased

Question 16

A researcher finds that 45 out of 54 patients have serum HDL between 34.0 and 69.1. What is the actual percentage?

Accepted Answer

$$\frac{45}{54} 	imes 100 \approx 83.3\%$$

Question 17

Which of the following is true about Chebyshev’s Inequality?

Accepted Answer

It applies to any data set, regardless of shape.

Question 18

For a bell-shaped IQ distribution with $$\mu = 100$$ and $$\sigma = 15$$, what percentage of people have IQ scores greater than 130?

Accepted Answer

2.5%

Question 19

What is the degrees of freedom when calculating the sample variance for a sample of size $$n$$?

Accepted Answer

$$n - 1$$

Measures of Dispersion: Range, Variance, Standard Deviation, Empirical Rule, and Chebyshev’s Inequality

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