Q: A contestant on a game show must shoot 8 shots at a target to win money. The contestant gets $$8$$ shots and each shot is successful with probability $$0.60. $$What is the standard deviation of $$X$$, the number of shots made?

$$\sqrt{8 \times 0.60 \times 0.40} = 1.386$$

Q: What is the standard error of $$\hat{p}$$ for the above scenario? (Round to 3 decimals)

$$\sqrt{\frac{0.70 \times 0.30}{10}} = 0.057$$

Question 1

Which of the following statements about the area under a continuous probability distribution is correct?

Accepted Answer

The total area under the curve is always equal to 1.

Question 2

A binomial distribution is characterized by which of the following properties?

Accepted Answer

A fixed number of independent trials, each with two possible outcomes.

Question 3

The standard error is best described as:

Accepted Answer

The variability of a sampling distribution.

Question 4

Which statement about the Central Limit Theorem is correct?

Accepted Answer

It states that the sampling distribution of the sample mean approaches normality as sample size increases.

Question 5

A z-score of 0 for an observation indicates:

Accepted Answer

The observation is equal to the mean.

Question 6

Suppose an unfair coin is flipped 3 times. The probability distribution for the number of heads observed is given as $$P(X=0)=0.10$$, $$P(X=1)=0.40$$, $$P(X=2)=0.40$$, $$P(X=3)=0.10. Is $$this a valid probability distribution?

Accepted Answer

No, because the sum of probabilities is not equal to 1.

Question 7

Given the probability distribution $$P(X=0)=0.10$$, $$P(X=1)=0.40$$, $$P(X=2)=0.40$$, what should $$P(X=3) be to $$make it a valid probability distribution?

Accepted Answer

$$P(X=3)=0.20$$

Question 8

Suppose an unfair coin is flipped 3 times and the probability distribution is $$P(Y=0)=0.10$$, $$P(Y=1)=0.41$$, $$P(Y=2)=0.41$$, $$P(Y=3)=0.08. $$What is the expected value $$E(Y)$$?

Accepted Answer

$$E(Y)=0.90$$

Question 9

Given the same unfair coin scenario, what is $$P(Y\u003c3)$$?

Accepted Answer

$$P(Y\u003c3)=0.93$$

Question 10

A mechanic has 20 unlabeled keys and only one opens a specific door. What is the probability the mechanic opens the door on the first try?

Accepted Answer

$$P=0.20$$

Question 11

Why does the scenario with the mechanic and 20 keys NOT follow a binomial distribution?

Accepted Answer

Because the probability of success changes after each trial.

Question 12

A snack food's calories follow a normal distribution with mean 120 and standard deviation 6. What is the z-score for a product with 110 calories?

Accepted Answer

$$z=-1.67$$

Question 13

Given the same normal distribution, what proportion of products have more than 112 calories? (Show calculation: $$z = \frac{112-120}{6})$$

Accepted Answer

$$P=0.0475$$

Question 14

Given the same normal distribution, what calorie value corresponds to the percentile of 90.147? (Show calculation: $$z=1.29)$$

Accepted Answer

$$126.45$$

Question 15

A contestant on a game show must shoot 8 shots at a target to win money. The contestant gets $$8$$ shots and each shot is successful with probability $$0.60. $$What is the standard deviation of $$X$$, the number of shots made?

Accepted Answer

$$\sqrt{8 	imes 0.60 	imes 0.40} = 1.386$$

Question 16

What is the probability the contestant makes exactly 5 shots? (Use binomial formula)

Accepted Answer

$$\binom{8}{5} (0.60)^5$ (0.40)^3$ = 0.277$$

Question 17

What is the probability the contestant makes at most 7 shots? (Use binomial formula)

Accepted Answer

$$1 - (0.60)^8$ = 0.9832$$

Question 18

After an election, 70% of voters voted for candidate A. Researchers conduct exit polls with random samples of size 10. What is the expected value of the sample proportion $$\hat{p}$$?

Accepted Answer

$$E(\hat{p}) = 0.70$$

Question 19

What is the standard error of $$\hat{p}$$ for the above scenario? (Round to 3 decimals)

Accepted Answer

$$\sqrt{\frac{0.70 	imes 0.30}{10}} = 0.057$$

Question 20

What is the probability researchers obtain a sample proportion in favor of candidate A less than 0.57? (Show calculation: $$z = \frac{0.57-0.70}{0.057})$$

Accepted Answer

$$P=0.3050$$

Probability Distributions, Sampling, and Statistical Inference: Study Notes

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