Q: Given a frequency distribution where $$x_i$$ values are 3, 5, and 7 with corresponding frequencies $$f_i of 2$$, 3, and 5, what is the arithmetic mean?

$$\frac{2 \times 3 + 3 \times 5 + 5 \times 7}{2 + 3 + 5} = \frac{6 + 15 + 35}{10} = \frac{56}{10} = 5.6$$

Q: A biologist measures the growth rates of three bacteria samples: 2, 8, and 32. What is the geometric mean of these growth rates?

$$(2 \times 8 \times 32$$)^{1/3}$$ = (512$$)^{1/3}$$ = 8$$

Q: A company wants to find the average growth rate of its investments over 4 years, with annual growth rates of 5%, 10%, 20%, and 40%. Which mean should be used and what is its value?

Geometric mean; $$G = (1.05 \times 1.10 \times 1.20 \times 1.40$$)^{1/4}$$ \approx 1.179 or 17.9$$%

Question 1

A researcher collects the following data on the number of books read by five students in a month: 2, 4, 6, 8, and 10. What is the arithmetic mean of the number of books read?

Accepted Answer

$$\frac{2 + 4 + 6 + 8 + 10}{5} = 6$$

Question 2

Given a frequency distribution where $$x_i$$ values are 3, 5, and 7 with corresponding frequencies $$f_i of 2$$, 3, and 5, what is the arithmetic mean?

Accepted Answer

$$\frac{2 	imes 3 + 3 	imes 5 + 5 	imes 7}{2 + 3 + 5} = \frac{6 + 15 + 35}{10} = \frac{56}{10} = 5.6$$

Question 3

Which of the following formulas correctly represents the geometric mean $$G of$$ $$n$$ non-zero observations $$x_1$$, $$x_2$$, ..., x_n$$?

Accepted Answer

$$G = (x_1$ x_2$ x_3$ ... x_n$$)^{1/n}$$

Question 4

A biologist measures the growth rates of three bacteria samples: 2, 8, and 32. What is the geometric mean of these growth rates?

Accepted Answer

$$(2 	imes 8 	imes 32$$)^{1/3}$$ = (512$$)^{1/3}$$ = 8$$

Question 5

Which statement best describes the harmonic mean of a set of non-zero observations $$x_1$$, $$x_2$$, ..., x_n$$?

Accepted Answer

It is the reciprocal of the arithmetic mean of the reciprocals of the observations.

Question 6

A car travels three segments at speeds of 30 km/h, 60 km/h, and 90 km/h. What is the harmonic mean of the speeds?

Accepted Answer

$$H = \frac{3}{\frac{1}{30} + \frac{1}{60} + \frac{1}{90}} = \frac{3}{\frac{1}{30} + \frac{1}{60} + \frac{1}{90}} = \frac{3}{\frac{6 + 3 + 2}{180}} = \frac{3}{\frac{11}{180}} = \frac{3 	imes 180}{11} \approx 49.09$$

Question 7

In a frequency distribution, if $$x_i$$ are the values and $$f_i$$ are their frequencies, which formula gives the arithmetic mean?

Accepted Answer

$$\bar{x} = \frac{\sum_{i=1}^{n} f_i x_i}{\sum_{i=1}^{n} f_i}$$

Question 8

A student records the following scores in five tests: 10, 20, 30, 40, and 50. What is the harmonic mean of the scores?

Accepted Answer

$$H = \frac{5}{\frac{1}{10} + \frac{1}{20} + \frac{1}{30} + \frac{1}{40} + \frac{1}{50}} \approx \frac{5}{0.2283} \approx 21.89$$

Question 9

Which measure of central tendency is most appropriate for averaging rates, such as speed or efficiency?

Accepted Answer

Harmonic mean

Question 10

If the arithmetic mean of a set of numbers is 12, what is the sum of the numbers if there are 8 observations?

Accepted Answer

$$12 	imes 8 = 96$$

Question 11

A company wants to find the average growth rate of its investments over 4 years, with annual growth rates of 5%, 10%, 20%, and 40%. Which mean should be used and what is its value?

Accepted Answer

Geometric mean; $$G = (1.05 	imes 1.10 	imes 1.20 	imes 1.40$$)^{1/4}$$ \approx 1.179 or 17.9$$%

Question 12

Which of the following is NOT a property of the arithmetic mean?

Accepted Answer

It is always less than the geometric mean for positive numbers.

Question 13

If the geometric mean of three numbers is 10 and two of the numbers are 5 and 20, what is the third number?

Accepted Answer

$$x = \frac{10^3$}{5 	imes 20} = \frac{1000}{100} = 10$$

Question 14

A dataset contains the values 1, 2, 4, and 8. Which mean will be the smallest?

Accepted Answer

Harmonic mean

Question 15

In a frequency distribution, if $$f_1 = 4$, x_1$ = 10$$; $$f_2 = 6$, x_2$ = 20$$; $$f_3 = 10$, x_3$ = 30$$, what is the arithmetic mean?

Accepted Answer

$$\bar{x} = \frac{4 	imes 10 + 6 	imes 20 + 10 	imes 30}{4 + 6 + 10} = \frac{40 + 120 + 300}{20} = \frac{460}{20} = 23$$

Question 16

Which mean is most appropriate for calculating average rates when the time or distance is constant?

Accepted Answer

Harmonic mean

Question 17

If the harmonic mean of two numbers is 4 and one of the numbers is 8, what is the other number?

Accepted Answer

$$x = \frac{2 	imes 8 	imes 4}{8 + 4} = \frac{64}{12} = 5.33$$

Question 18

A dataset contains the values 2, 4, 8, and 16. What is the geometric mean?

Accepted Answer

$$(2 	imes 4 	imes 8 	imes 16$$)^{1/4}$$ = (1024$$)^{1/4}$$ = 5.656$$

Statistics - By Ron

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