Q: For the same scenario as above, what is the standard deviation of the number of students attending the school? (Use $$p=0.62$$, $$q=0.38$$, $$n=500)$$

$$\sqrt{500 \times 0.62 \times 0.38}$$

Question 1

Given the probability distribution for the number of rotten tomatoes in a bag: $$P(X=0)=0.95$$, $$P(X=1)=0.02$$, $$P(X=2)=0.02$$, $$P(X=3)=0.01$$, what is the expected number of rotten tomatoes per bag?

Accepted Answer

$$0.09$$

Question 2

A raffle ticket costs $$10 and the prize is 15$$,000. If 2,000 tickets are sold, what is the expected value of buying one ticket?

Accepted Answer

$$-\2.50$

Question 3

Which of the following best describes how to calculate the expected value $$E(X)$$ for a discrete random variable $$X$$ with possible values $$x_i$$ and probabilities $$P(x_i)$$?

Accepted Answer

$$E(X) = \sum x_i \cdot P(x_i)$$

Question 4

If you have a probability distribution with $$P(X=0)=0.7$$, $$P(X=1)=0.2$$, $$P(X=2)=0.1$$, what is the mean of $$X$$?

Accepted Answer

$$0.9$$

Question 5

A game costs $$5 to $$play. You have a $$1/100$$ chance to win $$400$$, and a $$99/100$$ chance to win nothing. What is the expected value of playing the game?

Accepted Answer

Question 6

Suppose 62% of 12th graders attend a particular school. In a sample of 500 students, what is the mean number of students expected to attend the school?

Accepted Answer

$$310$$

Question 7

For the same scenario as above, what is the standard deviation of the number of students attending the school? (Use $$p=0.62$$, $$q=0.38$$, $$n=500)$$

Accepted Answer

$$\sqrt{500 	imes 0.62 	imes 0.38}$$

Question 8

In a binomial distribution, when can you use the normal approximation?

Accepted Answer

When both $$np \geq 5$$ and $$nq \geq 5$$

Question 9

What is the continuity correction factor when approximating a binomial probability $$P(X \geq k)$$ with a normal distribution?

Accepted Answer

Use $$P(X \geq k - 0.5)$$

Question 10

If the mean of a binomial distribution is $$\mu = np$$ and the standard deviation is $$\sigma = \sqrt{npq}$$, what are $$p$$ and $$q$$?

Accepted Answer

$$p is $$the probability of success, $$q = 1 - p$$

Question 11

A lottery ticket costs $$2. $$The prize is $$1,000$$ and there are 1,000 tickets. What is the expected value of buying one ticket?

Accepted Answer

$$-\1.00$

Question 12

If $$P(X=0)=0.8$$, $$P(X=1)=0.15$$, $$P(X=2)=0.05$$, what is the variance of $$X$$?

Accepted Answer

$$0.175$$

Question 13

A sample of 500 students is taken, and the probability of a student attending a certain school is $$0.62. $$What is the probability that at least 290 students attend, using the normal approximation and continuity correction?

Accepted Answer

$$P(X \u003e 289.5)$$

Question 14

In the context of expected value, which of the following statements is true?

Accepted Answer

Expected value represents the long-run average outcome of a random process.

Question 15

If you have a probability distribution with $$E(X) = 0.09$$ and you play the game 100 times, what is your expected total outcome?

Accepted Answer

$$9$$

Question 16

A binomial random variable $$X$$ has $$n=500$$, $$p=0.62. $$What is the $$z-$$score for $$X=289.5$$?

Accepted Answer

$$z = \frac{289.5 - 310}{10.85}$$

Question 17

Which of the following is NOT a step in finding the expected value for a probability distribution?

Accepted Answer

Divide each value by its probability

Question 18

If the expected value of a raffle ticket is negative, what does this imply for a rational player?

Accepted Answer

They should not buy the ticket, as they expect to lose money in the long run.

Probability Distributions and Expected Value in Statistics

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