Q: A random sample of 200 Americans is asked if they think it is important to control gun ownership ($$p = 0.47). $$What is the expected number of people who say 'yes'?

$$n \times p = 200 \times 0.47 = 94$$

Q: Suppose a baseball player has a batting average of 0.308 and bats 480 times in a season. What is the standard deviation of the number of hits?

$$\sqrt{480 \times 0.308 \times (1 - 0.308)}$$

Q: If $$X is a $$binomial random variable with $$n = 15$$ and $$p = 0.3$$, what is the mean of $$X$$?

$$n \times p = 15 \times 0.3 = 4.5$$

Q: A random variable $$X$$ follows a Normal distribution with mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What value of $$X$$ corresponds to the 97.5th percentile? (Use $$z = 1.96$$ for the 97.5th percentile.)

$$X = 100 + 1.96 \times 15 = 129.4$$

Question 1

Which of the following is a key feature of a probability distribution for a discrete random variable?

Accepted Answer

All probabilities must be between 0 and 1, and their sum must equal 1.

Question 2

Suppose you roll a fair 6-sided die. You win $$10 if you roll a 5 or a 6, lose 5 if $$you roll a 1, and win/lose nothing otherwise. What is the probability that you win $10?

Accepted Answer

$$\frac{1}{3}$$

Question 3

Which of the following variables is continuous?

Accepted Answer

Height of a building in downtown San Francisco

Question 4

A probability density curve for a continuous random variable must satisfy which of the following?

Accepted Answer

The total area under the curve must be 1.

Question 5

A bus arrives every 12 minutes. What is the probability that a randomly arriving passenger waits between 4 and 10 minutes, assuming a uniform distribution?

Accepted Answer

$$\frac{6}{12}$$

Question 6

Which of the following best describes the Normal model?

Accepted Answer

A probability model for continuous random variables with a bell-shaped, symmetric curve

Question 7

Given a Normal distribution $$N(64, 3)$$, what is the z-score for a value of 67?

Accepted Answer

$$z = 1$$

Question 8

Suppose the birth lengths of seal pups are normally distributed with mean $$\mu = 29.5$$ inches and standard deviation $$\sigma = 1.2$$ inches. What is the probability that a randomly selected pup is shorter than 28.0 inches? (Find the z-score first.)

Accepted Answer

$$z = -1.25$$

Question 9

Which of the following is NOT a characteristic of a binomial probability model?

Accepted Answer

The trials are dependent.

Question 10

A random sample of 200 Americans is asked if they think it is important to control gun ownership ($$p = 0.47). $$What is the expected number of people who say 'yes'?

Accepted Answer

$$n 	imes p = 200 	imes 0.47 = 94$$

Question 11

Suppose a baseball player has a batting average of 0.308 and bats 480 times in a season. What is the standard deviation of the number of hits?

Accepted Answer

$$\sqrt{480 	imes 0.308 	imes (1 - 0.308)}$$

Question 12

Which of the following scenarios is NOT binomial?

Accepted Answer

Recording the ages of 20 randomly selected students

Question 13

If $$X is a $$binomial random variable with $$n = 15$$ and $$p = 0.3$$, what is the mean of $$X$$?

Accepted Answer

$$n 	imes p = 15 	imes 0.3 = 4.5$$

Question 14

A random variable $$X$$ follows a Normal distribution with mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What value of $$X$$ corresponds to the 97.5th percentile? (Use $$z = 1.96$$ for the 97.5th percentile.)

Accepted Answer

$$X = 100 + 1.96 	imes 15 = 129.4$$

Question 15

Suppose the probability of being unemployed in a city is $$p = 0.102. In a $$random sample of $$n = 15$$ people, what is the probability that three or fewer are unemployed? (Express as a cumulative binomial probability.)

Accepted Answer

$$P(X \leq 3)$$

Question 16

Which of the following statements about the shape of a binomial distribution is correct?

Accepted Answer

It is symmetric only if $$p = 0.5 or$$ $$n is $$large.

Question 17

A random variable $$X is $$normally distributed with mean $$\mu = 50$$ and standard deviation $$\sigma = 5. $$What is the probability that $$X is $$greater than 60? (Find the z-score first.)

Accepted Answer

$$z = 2$$

Question 18

In a binomial experiment with $$n = 80$$ and $$p = 0.11$$, what is the probability that more than 15 people lack medical insurance?

Accepted Answer

$$P(X \u003e 15)$$

Question 19

Which of the following best describes the difference between the Normal and Binomial distributions regarding the probability of 'more than 5' versus '5 or more'?

Accepted Answer

For the Normal distribution, $$P(X \u003e 5) = P(X \geq 5)$$, but for the Binomial distribution, $$P(X \u003e 5) 
eq P(X \geq 5).$$

Modeling Random Events: The Normal and Binomial Models – Study Notes

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