Q: A survey of 115 people about the number of TVs in their house yields the following frequencies: 0 TVs (9), 1 TV (25), 2 TVs (29), 3 TVs (19), 4 TVs (15), 5 TVs (8), 6 TVs (6), 7 TVs (4). What is the mean number of TVs per household?

$$\frac{(0 \times 9) + (1 \times 25) + (2 \times 29) + (3 \times 19) + (4 \times 15) + (5 \times 8) + (6 \times 6) + (7 \times 4)}{115} = 2.51$$

Q: In a binomial experiment where 84% of American high school students graduate, and 10 students are sampled, what is the probability that exactly 3 students graduated?

$$\binom{10}{3} \times (0.84)^3$ \times (0.16)^7$

Q: For the same binomial experiment, what is the mean number of students who graduated in a sample of 10?

$$\mu = n \times p = 10 \times 0.84 = 8.4$$

Q: An elementary school raffle sells 1,000 tickets for $$1 each. The winner gets 500$$, everyone else gets nothing. What is the expected value for a person who buys one ticket?

$$E = (\frac{1}{1000} \times 500) + (\frac{999}{1000} \times 0) - 1 = -0.50$$

Q: In roulette, betting $$1 on a number has a probability of winning $$\frac{1}{38}$$ and losing $$\frac{37}{38}$$. The payoff is 35. $$What is the expected payoff?

$$E = (\frac{1}{38} \times 35) + (\frac{37}{38} \times -1) = -0.0526$$

Q: To qualify for Calculus, a student must score in the top 15% on a placement test with mean $$50$$ and standard deviation $$4. $$What is the lowest possible qualifying score?

$$x = 50 + (z_{0.85} \times 4)$$,\ z_{0.85} \approx 1.036$$;\ x = 50 + (1.036$$ \times 4) = 54.14$

Question 1

A single card is drawn from a standard deck of 52 cards. What is the probability that the card is Red or Diamond?

Accepted Answer

$$\frac{39}{52}$$

Question 2

At a used-book sale, there are 100 adult books (70 nonfiction) and 160 children’s books (100 fiction). What is the probability that a randomly selected book is a fiction book?

Accepted Answer

$$\frac{130}{260}$$

Question 3

At the same book sale, what is the probability that a randomly selected book is a nonfiction children’s book?

Accepted Answer

$$\frac{60}{260}$$

Question 4

At the book sale, what is the probability that a randomly selected book is either a fiction book or an adult book?

Accepted Answer

$$\frac{190}{260}$$

Question 5

A survey of 115 people about the number of TVs in their house yields the following frequencies: 0 TVs (9), 1 TV (25), 2 TVs (29), 3 TVs (19), 4 TVs (15), 5 TVs (8), 6 TVs (6), 7 TVs (4). What is the mean number of TVs per household?

Accepted Answer

$$\frac{(0 	imes 9) + (1 	imes 25) + (2 	imes 29) + (3 	imes 19) + (4 	imes 15) + (5 	imes 8) + (6 	imes 6) + (7 	imes 4)}{115} = 2.51$$

Question 6

In a binomial experiment where 84% of American high school students graduate, and 10 students are sampled, what is the probability that exactly 3 students graduated?

Accepted Answer

$$\binom{10}{3} 	imes (0.84)^3$ 	imes (0.16)^7$

Question 7

For the same binomial experiment, what is the mean number of students who graduated in a sample of 10?

Accepted Answer

$$\mu = n 	imes p = 10 	imes 0.84 = 8.4$$

Question 8

An elementary school raffle sells 1,000 tickets for $$1 each. The winner gets 500$$, everyone else gets nothing. What is the expected value for a person who buys one ticket?

Accepted Answer

$$E = (\frac{1}{1000} 	imes 500) + (\frac{999}{1000} 	imes 0) - 1 = -0.50$$

Question 9

In roulette, betting $$1 on a number has a probability of winning $$\frac{1}{38}$$ and losing $$\frac{37}{38}$$. The payoff is 35. $$What is the expected payoff?

Accepted Answer

$$E = (\frac{1}{38} 	imes 35) + (\frac{37}{38} 	imes -1) = -0.0526$$

Question 10

Replacement times for TV sets are normally distributed with mean $$8.2$$ years and standard deviation $$1.1$$ years. What is the probability that a randomly selected TV will be replaced within the first 5 years?

Accepted Answer

$$P(X \u003c 5) = P\left(Z \u003c \frac{5 - 8.2}{1.1}\right) = P(Z \u003c -2.91)$$

Question 11

To qualify for Calculus, a student must score in the top 15% on a placement test with mean $$50$$ and standard deviation $$4. $$What is the lowest possible qualifying score?

Accepted Answer

$$x = 50 + (z_{0.85} 	imes 4)$$,\ z_{0.85} \approx 1.036$$;\ x = 50 + (1.036$$ 	imes 4) = 54.14$

Question 12

The weights of 1-year-old boys are normally distributed with mean $$22.8$$ lbs and standard deviation $$2.15$$ lbs. What is the standard deviation of the sampling distribution of the sample mean for a sample of 10 boys?

Accepted Answer

$$\sigma_{\bar{x}} = \frac{2.15}{\sqrt{10}} = 0.68$$

Question 13

For the same sample of 10 boys, what is the probability that the mean weight is greater than 21 lbs?

Accepted Answer

$$P(\bar{x} \u003e 21) = P\left(Z \u003e \frac{21 - 22.8}{0.68}\right) = P(Z \u003e -2.65)$$

Question 14

According to CreditCards.com, 29% of adults do not own a credit card. In a random sample of 500 adults, what is the mean and standard deviation of the sampling distribution of the sample proportion?

Accepted Answer

$$\mu_{\hat{p}} = 0.29$$,\ \sigma_{\hat{p}} = \sqrt{\frac{0.29 	imes 0.71}{500}} = 0.0202$

Question 15

In the same credit card survey, what is the probability that between 25% and 30% of the sample do not own a credit card?

Accepted Answer

$$P(0.25 \u003c \hat{p} \u003c 0.30) = P\left(\frac{0.25 - 0.29}{0.0202} \u003c Z \u003c \frac{0.30 - 0.29}{0.0202}\right) = P(-1.98 \u003c Z \u003c 0.50)$$

Question 16

A random sample of 500 adults is surveyed about credit card ownership. Would it be unusual for 125 or fewer adults to not own a credit card? Why?

Accepted Answer

Yes, because $$\hat{p} = \frac{125}{500} = 0.25 is $$more than 2 standard deviations below the mean $$0.29.$$

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