Q: If $$n = 460$$ and $$p = 0.5$$, what is the mean ($$\mu) of $$the binomial distribution?

$$\mu = n p = 460 \times 0.5 = 230$$

Q: What is the standard deviation ($$\sigma) $$for a binomial distribution with $$n = 460$$ and $$p = 0.5$$?

$$\sigma = \sqrt{n p q} = \sqrt{460 \times 0.5 \times 0.5}$$

Question 1

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 2

Given $$n = 10$$, $$p = 0.3$$, and $$q = 0.7$$, what does $$q$$ represent in the context of a binomial experiment?

Accepted Answer

The probability of failure in any one trial.

Question 3

Which formula correctly represents the probability of exactly $$x$$ successes in $$n$$ binomial trials?

Accepted Answer

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

Question 4

In a clinical trial, a new drug has a 70% chance of success per patient. If 5 patients are treated, what is the probability that exactly 3 will be successful? Use $$p = 0.7$$, $$q = 0.3$$, $$n = 5$$, $$x = 3.$$

Accepted Answer

$$P(X = 3) = \frac{5!}{3!2!} (0.7)^3$ (0.3)^2$

Question 5

If $$n = 460$$ and $$p = 0.5$$, what is the mean ($$\mu) of $$the binomial distribution?

Accepted Answer

$$\mu = n p = 460 	imes 0.5 = 230$$

Question 6

What is the standard deviation ($$\sigma) $$for a binomial distribution with $$n = 460$$ and $$p = 0.5$$?

Accepted Answer

$$\sigma = \sqrt{n p q} = \sqrt{460 	imes 0.5 	imes 0.5}$$

Question 7

A fair coin is flipped 8 times. What is the probability of getting exactly 5 heads?

Accepted Answer

$$P(X = 5) = \frac{8!}{5!3!} (0.5)^5$ (0.5)^3$

Question 8

Which of the following scenarios can be modeled using a binomial distribution?

Accepted Answer

Counting the number of defective items in a batch of 100, where each item is either defective or not.

Question 9

If the probability of success in a binomial experiment increases, what happens to the mean of the distribution, assuming $$n is $$constant?

Accepted Answer

The mean increases.

Question 10

Using the range rule of thumb, what is the range of values that are not considered significantly low or high for the number of wins in 460 NFL overtime games, with $$p = 0.5$$?

Accepted Answer

$$\mu \pm 2\sigma$$, where $$\mu = 230$$ and $$\sigma = \sqrt{460 	imes 0.5 	imes 0.5}$$

Question 11

A researcher claims that getting 252 wins in 460 NFL overtime games is significantly high. Using the mean and standard deviation, how would you evaluate this claim?

Accepted Answer

Calculate if 252 is greater than $$\mu + 2\sigma$$; if so, it is significantly high.

Question 12

Which of the following best describes the meaning of 'independent trials' in a binomial experiment?

Accepted Answer

The outcome of one trial does not affect the probability of outcomes in other trials.

Question 13

If $$p = 0.2$$ and $$n = 15$$, what is the probability of getting zero successes?

Accepted Answer

$$P(X = 0) = (0.8$$)^{15}$$

Question 14

A quality control inspector checks 20 products, each with a 5% chance of being defective. What is the expected number of defective products?

Accepted Answer

$$\mu = n p = 20 	imes 0.05 = 1$$

Question 15

Which of the following is the correct formula for $$q in a $$binomial distribution?

Accepted Answer

$$q = 1 - p$$

Binomial Probability Distributions: Concepts, Notation, and Applications

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