Q: If $$n = 10$$, $$p = 0.05$$, and $$q = 0.95$$, what is the standard deviation ($$\sigma) of $$the binomial distribution?

$$\sigma = \sqrt{10 \times 0.05 \times 0.95}$$

Q: In the NFL overtime example, $$n = 460$$ and $$p = 0.5. $$What is the variance ($$\$$sigma^2$$) of the number of wins?

$$\$$sigma^2$$ = 460 \times 0.5 \times 0.5 = 115$$

Q: Using the range rule of thumb, for a binomial distribution with $$\mu = 230$$ and $$\sigma = 10.7$$, what is the minimum value considered significantly high?

$$230 + 2 \times 10.7 = 251.4$$

Q: A researcher wants to know if getting 252 wins in 460 NFL overtime games is significantly high. If $$p = 0.5$$, $$n = 460$$, and $$\sigma = 10.7$$, which calculation would you use to check if 252 is significantly high using the range rule of thumb?

Check if $$252 \geq 230 + 2 \times 10.7$$

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

In a clinical trial, a new drug is tested on 20 patients. Each patient either responds to the drug or does not. The probability that a patient responds is 0.3. What is the probability that exactly 5 patients respond to the drug? Use the binomial probability formula.

Accepted Answer

$$P(5) = \frac{20!}{15!5!} \cdot 0.3^5$ \cdot 0.7$$^{15}$$

Question 3

Given a binomial distribution with $$n = 12$$, $$p = 0.4$$, and $$q = 0.6$$, what is the mean ($$\mu) of $$the distribution?

Accepted Answer

$$\mu = 12 	imes 0.4 = 4.8$$

Question 4

A taste test is conducted where each participant chooses between two colas. If 8 people participate and the probability of preferring Cola A is 0.5, what is the probability that exactly 3 people prefer Cola A?

Accepted Answer

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

Question 5

If $$n = 10$$, $$p = 0.05$$, and $$q = 0.95$$, what is the standard deviation ($$\sigma) of $$the binomial distribution?

Accepted Answer

$$\sigma = \sqrt{10 	imes 0.05 	imes 0.95}$$

Question 6

A car salesperson has a 20% chance of making a sale to each customer. If she sees 15 customers in a week, what is the expected number of sales?

Accepted Answer

$$\mu = 15 	imes 0.2 = 3$$

Question 7

Which of the following scenarios does NOT result in a binomial distribution?

Accepted Answer

Measuring the time until the first success in a series of trials.

Question 8

In the NFL overtime example, $$n = 460$$ and $$p = 0.5. $$What is the variance ($$\$$sigma^2$$) of the number of wins?

Accepted Answer

$$\$$sigma^2$$ = 460 	imes 0.5 	imes 0.5 = 115$$

Question 9

Using the range rule of thumb, for a binomial distribution with $$\mu = 230$$ and $$\sigma = 10.7$$, what is the minimum value considered significantly high?

Accepted Answer

$$230 + 2 	imes 10.7 = 251.4$$

Question 10

A survey finds that 5% of adults are cashless. If 10 adults are randomly selected, what is the probability that none are cashless? Use the binomial probability formula.

Accepted Answer

$$P(0) = \frac{10!}{10!0!} \cdot 0.05^0$ \cdot 0.95$$^{10}$$

Question 11

If the probability of a success in a binomial experiment is $$p$$, what is the probability of a failure?

Accepted Answer

$$q = 1 - p$$

Question 12

A researcher wants to know if getting 252 wins in 460 NFL overtime games is significantly high. If $$p = 0.5$$, $$n = 460$$, and $$\sigma = 10.7$$, which calculation would you use to check if 252 is significantly high using the range rule of thumb?

Accepted Answer

Check if $$252 \geq 230 + 2 	imes 10.7$$

Question 13

Which Excel function would you use to find the probability of exactly 2 successes in 10 trials with a success probability of 0.05?

Accepted Answer

=BINOM.DIST(2,10,0.05,0)

Question 14

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

Accepted Answer

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

Question 15

A binomial experiment has $$n = 15$$ and $$p = 0.2. $$What is the probability of getting exactly 0 successes?

Accepted Answer

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

Question 16

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.

5.2 Binomial Probability Distributions: Concepts, Calculations, and Applications

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