Question 1

Which of the following scenarios best fits the requirements for a binomial probability distribution?

Accepted Answer

Counting the number of defective light bulbs in a sample of 20, where each bulb is tested and replaced before the next is tested, and the probability of a defect is constant.

Question 2

A pharmaceutical company tests a new drug on 8 patients. Each patient has a 0.25 probability of experiencing a side effect, independent of the others. What is the probability that exactly 3 patients experience the side effect? Use the binomial probability formula.

Accepted Answer

$$P(3) = \dfrac{8!}{5!3!} \cdot 0.25^3$ \cdot 0.75^5$ = 56 \cdot 0.015625 \cdot 0.237305 = 0.208$$

Question 3

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

Accepted Answer

Each trial must have more than two possible outcomes.

Question 4

Suppose 5% of U.S. adults are vegetarians. If you randomly select 5 adults, what is the probability that exactly 2 are vegetarians? (Use the binomial probability formula.)

Accepted Answer

$$P(2) = \dfrac{5!}{3!2!} \cdot 0.05^2$ \cdot 0.95^3$ = 10 \cdot 0.0025 \cdot 0.857375 = 0.021$$

Question 5

A quality control manager samples 100 products from a large shipment (over 10,000 items) without replacement. If the defective rate is 2%, which guideline allows her to treat the trials as independent?

Accepted Answer

The 5% guideline for cumbersome calculations.

Question 6

Given a binomial distribution with $$n = 20$$ and $$p = 0.3$$, what are the mean and standard deviation?

Accepted Answer

Mean: $$\mu = np = 6$$; Standard deviation: $$\sigma = \sqrt{npq} = \sqrt{20 	imes 0.3 	imes 0.7} = 2.05$$

Question 7

In a clinical trial, the probability that a patient recovers from a disease with a new treatment is 0.8. If 12 patients are treated, what is the probability that at least 10 recover? (Select the correct approach.)

Accepted Answer

Sum the probabilities $$P(10) + P(11) + P(12)$$ using the binomial formula.

Question 8

A survey finds that 10% of people are left-handed. In a random sample of 15 people, what is the probability that no more than 2 are left-handed?

Accepted Answer

$$P(0) + P(1) + P(2)$$, where $$P(x) = \dfrac{15!}{x!(15-x)!} \cdot 0.1^x \cdot 0.9$$^{15-x}$$

Question 9

In a binomial experiment with $$n = 460$$ and $$p = 0.5$$, what is the range of values that are NOT considered significantly high or low according to the range rule of thumb?

Accepted Answer

Between $$\mu - 2\sigma$$ and $$\mu + 2\sigma$$, or between $$230.0 - 2(10.7) = 208.6$$ and $$230.0 + 2(10.7) = 251.4$$

Question 10

A researcher claims that a coin is biased. In 20 flips, the coin lands heads 17 times. Assuming the coin is fair, is this result significantly high? (Use $$p = 0.5$$ and the range rule of thumb.)

Accepted Answer

Yes, because $$17 \u003e \mu + 2\sigma$$ where $$\mu = 10$$, $$\sigma = \sqrt{20 	imes 0.5 	imes 0.5} = 2.24$$, so $$\mu + 2\sigma = 14.48.$$

Question 11

Which of the following best describes the meaning of 'success' in a binomial probability distribution?

Accepted Answer

It is an arbitrary label for one of the two possible outcomes, not necessarily something good.

Question 12

If $$P(x 	ext{ or more}) \u003c 0.05 in a $$binomial experiment, what can you conclude about $$x$$?

Accepted Answer

$$x is a $$significantly high number of successes.

Question 13

A hospital administrator wants to know if getting 8 or more infections in 50 surgeries (with infection probability $$p = 0.05) is $$significantly high. Which calculation should she perform?

Accepted Answer

Calculate $$P(x \geq 8)$$ and check if it is less than 0.05.

Question 14

Which formula gives the probability of exactly $$x$$ successes in $$n$$ binomial trials with probability $$p of $$success?

Accepted Answer

$$P(x) = \dfrac{n!}{(n-x)!x!} p^x q$$^{n-x}$$

Question 15

A company finds that 30% of its employees work remotely. If 12 employees are randomly selected, what is the expected number of remote workers and the variance?

Accepted Answer

Expected: $$\mu = np = 12 	imes 0.3 = 3.6$$; Variance: $$\$$sigma^2$$ = npq = 12 	imes 0.3 	imes 0.7 = 2.52$$

Question 16

A binomial experiment has $$n = 10$$ and $$p = 0.05. $$What is the probability that exactly 2 successes occur? (Round to four decimal places.)

Accepted Answer

$$P(2) = \dfrac{10!}{8!2!} \cdot 0.05^2$ \cdot 0.95^8$ = 45 \cdot 0.0025 \cdot 0.663420 = 0.0746$$

Question 17

If a binomial probability distribution has $$n = 100$$ and $$p = 0.1$$, which value is considered significantly low according to the range rule of thumb?

Accepted Answer

Any value $$\leq \mu - 2\sigma$$, where $$\mu = 10$$ and $$\sigma = \sqrt{100 	imes 0.1 	imes 0.9} = 3$$; so, $$10 - 2(3) = 4$$

Binomial Probability Distributions: Concepts, Notation, and Applications

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