Q: A hospital records the blood types of 100 patients. If 40 have type O, what percentile is a patient with type O?

$$\frac{\text{number of values less than O}}{100} \cdot 100$$

Q: If events A and B are independent, which formula gives $$P(A \text{ and } B)$$?

$$P(A \text{ and } B) = P(A) \cdot P(B)$$

Question 1

A researcher collects the following data: 2, 4, 6, 8, 10. What is the midrange of this data set?

Accepted Answer

$$\frac{2+10}{2} = 6$$

Question 2

Which formula would you use to calculate the mean from a frequency table?

Accepted Answer

$$\frac{\sum f \cdot x}{\sum f}$$

Question 3

A student scored 85 on a test where the mean was 80 and the standard deviation was 5. What is the student's z-score?

Accepted Answer

$$z = \frac{85 - 80}{5} = 1$$

Question 4

In a binomial experiment with $$n = 10$$ trials and probability of success $$p = 0.3$$, what is the mean number of successes?

Accepted Answer

$$\mu = n \cdot p = 10 \cdot 0.3 = 3$$

Question 5

A hospital records the blood types of 100 patients. If 40 have type O, what percentile is a patient with type O?

Accepted Answer

$$\frac{	ext{number of values less than O}}{100} \cdot 100$$

Question 6

If events A and B are independent, which formula gives $$P(A 	ext{ and } B)$$?

Accepted Answer

$$P(A 	ext{ and } B) = P(A) \cdot P(B)$$

Question 7

A bag contains 5 red and 7 blue marbles. If two marbles are drawn without replacement, what is the probability both are red?

Accepted Answer

$$\frac{5}{12} \cdot \frac{4}{11} = \frac{20}{132} = \frac{5}{33}$$

Question 8

Which formula calculates the variance of a binomial distribution?

Accepted Answer

$$\$$sigma^2$$ = n \cdot p \cdot q$$

Question 9

A survey finds that 60% of people prefer tea over coffee. If 200 people are surveyed, what is the expected number who prefer tea?

Accepted Answer

$$E = n \cdot p = 200 \cdot 0.6 = 120$$

Question 10

A student wants to estimate the proportion of students who own laptops with a margin of error $$E = 0.05$$ and $$z_{\alpha/2} = 1.96. $$What is the minimum sample size required if no prior estimate is available?

Accepted Answer

$$n = \left[1.96\right]^2 \cdot 0.25 / 0.05^2$ = 384.16$$

Question 11

Which formula is used to calculate the test statistic for a population mean when the population standard deviation $$\sigma is $$known?

Accepted Answer

$$z = \frac{x - \mu}{\sigma / \sqrt{n}}$$

Question 12

A company wants to construct a 95% confidence interval for the mean weight of packages shipped, with $$\sigma = 2 kg$$, sample mean $$\bar{x} = 10 kg$$, and $$n = 25. $$What is the margin of error?

Accepted Answer

$$E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} = 1.96 \cdot \frac{2}{5} = 0.784$$

Question 13

Which formula is used to calculate the number of ways to choose 3 students from a group of 10?

Accepted Answer

$$nCr = \frac{n!}{(n-r)!r!} = \frac{10!}{7!3!}$$

Question 14

A discrete probability distribution has $$P(x)$$ values for $$x = 1, 2, 3. $$Which formula calculates the mean?

Accepted Answer

$$\mu = \sum x \cdot P(x)$$

Question 15

A regression analysis yields the equation $$y = 2x + 5. $$What is the predicted value of $$y$$ when $$x = 3$$?

Accepted Answer

$$y = 2 \cdot 3 + 5 = 11$$

Question 16

Which formula is used to calculate the standard deviation of a sample from a frequency table?

Accepted Answer

$$s = \sqrt{\frac{n \sum f \cdot x^2$ - (\sum f \cdot x)^2$}{n(n-1)}}$$

Question 17

A goodness of fit test uses the formula $$\sum \frac{(O-E)^2$}{E}. $$What does $$E$$ represent?

Accepted Answer

The expected frequency for each category

Question 18

If the sample mean is $$\bar{x} = 50$$, sample standard deviation $$s = 10$$, and $$n = 25$$, what is the 95% confidence interval for the mean (unknown $$\sigma$$)?

Accepted Answer

$$50 \pm t_{\alpha/2} \cdot \frac{10}{5}$$

Question 19

A school wants to estimate the mean GPA with a margin of error $$E = 0.1$$, $$\sigma = 0.5$$, and $$z_{\alpha/2} = 1.96. $$What is the minimum sample size required?

Accepted Answer

$$n = \left(\frac{1.96 \cdot 0.5}{0.1}\$$right)^2$$ = 96.04$$

Essential Formulas and Concepts in Elementary Statistics

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