Q: A batch contains 1000 pacemakers, 834 of which are non-defective. If three are selected at random without replacement and all are non-defective, what is the probability the batch is accepted?

$$\left(\frac{834}{1000}\right) \times \left(\frac{833}{999}\right) \times \left(\frac{832}{998}\right)$$

Q: A sample of 40 voters includes 27 Democrats. If the probability of a Democrat is 0.5, is this result significantly high?

Yes, because $$P(27 \text{ or more}) \leq 0.05$$

Q: A batch contains 8 green and 2 yellow peas. If 4 peas are selected at random without replacement, what is the probability all are green?

$$\frac{8}{10} \times \frac{7}{9} \times \frac{6}{8} \times \frac{5}{7}$$

Q: A normal distribution has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What value separates the top 10% of the distribution?

$$100 + 1.2816 \times 15 = 119.22$$

Question 1

A clinical trial tests a new medication for headaches. In the treatment group, 26 out of 103 subjects report headaches, while in the control group, 22 out of 71 report headaches. What is the relative risk of headaches in the treatment group compared to the control group?

Accepted Answer

$$\frac{26/103}{22/71}$$

Question 2

A batch of 15 smartphones contains 20% defective units. If you randomly select 3 phones with replacement, what is the probability that at least one is defective?

Accepted Answer

$$1 - (0.8)^3$ = 0.488$$

Question 3

Which of the following scenarios best illustrates the use of the multiplication rule for independent events?

Accepted Answer

Calculating the probability of flipping two coins and both landing heads.

Question 4

A university claims that the mean body temperature of students is $$98.6^\circF. In a $$random sample of 50 students, the mean is $$98.2^\circF. If $$the probability of obtaining a sample mean of $$98.2^\circF or $$lower is $$0.0001$$, what does this suggest about the university's claim?

Accepted Answer

The claim is likely incorrect because the observed mean is significantly low.

Question 5

A lottery requires you to select 5 numbers from 1 to 50 in any order. Which counting rule should you use to determine the number of possible combinations?

Accepted Answer

Combinations: $$C(50,5) = \frac{50!}{5!\,45!}$$

Question 6

Suppose the probability that a randomly selected adult carries no cash is $$0.17. If $$you select 8 adults at random, what is the probability that none of them carry cash?

Accepted Answer

$$(0.17)^8$$

Question 7

A bone density test score is normally distributed with mean $$0$$ and standard deviation $$1. $$What is the probability that a randomly selected score is between $$-1$$ and $$1$$?

Accepted Answer

Approximately $$0.6827$$

Question 8

A batch contains 1000 pacemakers, 834 of which are non-defective. If three are selected at random without replacement and all are non-defective, what is the probability the batch is accepted?

Accepted Answer

$$\left(\frac{834}{1000}ight) 	imes \left(\frac{833}{999}ight) 	imes \left(\frac{832}{998}ight)$$

Question 9

If the probability of a car rolling over in a crash is $$0.016$$, what is the probability that in 5 independent crashes, at least one car rolls over?

Accepted Answer

$$1 - (1 - 0.016)^5$

Question 10

A multiple-choice test has 10 questions, each with 5 possible answers. What is the probability of getting at least one correct answer by guessing on all questions?

Accepted Answer

$$1 - (0.8$$)^{10}$$

Question 11

A population has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the standard deviation of the sample mean for samples of size $$n = 36$$?

Accepted Answer

$$\frac{15}{\sqrt{36}} = 2.5$$

Question 12

A normal distribution has mean $$\mu = 60$$ and standard deviation $$\sigma = 12. $$What is the $$z-$$score for a value $$x = 75$$?

Accepted Answer

$$z = \frac{75 - 60}{12} = 1.25$$

Question 13

A genetic disease is X-linked. If the mother is a carrier (genotype xX) and the father is normal (XY), what is the probability that a son will inherit the disease?

Accepted Answer

$$0.5$$

Question 14

Which of the following is NOT a requirement for a binomial probability distribution?

Accepted Answer

The probability of success changes from trial to trial.

Question 15

A roulette wheel has 18 red, 18 black, and 2 green slots. What is the probability of getting a red slot in a single spin?

Accepted Answer

$$\frac{18}{38}$$

Question 16

A sample of 40 voters includes 27 Democrats. If the probability of a Democrat is 0.5, is this result significantly high?

Accepted Answer

Yes, because $$P(27 	ext{ or more}) \leq 0.05$$

Question 17

If the area to the left of a $$z-$$score is $$0.9500$$, what is the value of $$z$$?

Accepted Answer

$$1.645$$

Question 18

A batch contains 8 green and 2 yellow peas. If 4 peas are selected at random without replacement, what is the probability all are green?

Accepted Answer

$$\frac{8}{10} 	imes \frac{7}{9} 	imes \frac{6}{8} 	imes \frac{5}{7}$$

Question 19

A normal distribution has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What value separates the top 10% of the distribution?

Accepted Answer

$$100 + 1.2816 	imes 15 = 119.22$$

Question 20

A sample of size $$n = 50 is $$drawn without replacement from a population of $$N = 275$$ with $$\sigma = 16.0. $$What is the finite population correction factor for the standard deviation of the sample mean?

Accepted Answer

$$\sqrt{\frac{N-n}{N-1}} = \sqrt{\frac{225}{274}}$$

Probability, Counting, and Normal Distributions: Structured Study Notes for Statistics Students

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