Q: You randomly sample 400 students and find that 72% support a new policy. What is the 95% confidence interval for the proportion of students who support the policy? (Use $$z^* = 1.96)$$

$$0.72 \pm 1.96 \times \sqrt{\frac{0.72 \times 0.28}{400}}$$

Q: In a probability distribution for a dice game, you win 10 points if the sum is less than 4, 3 points if the sum is 4, 5, or 6, and lose 2 points if the sum is 7. What is the expected value of your winnings per roll?

$$E = 10 \times \frac{3}{36} + 3 \times \frac{9}{36} - 2 \times \frac{6}{36}$$

Q: A survey of 2500 students finds that 75% plan to attend an event. What is the standard error for the proportion?

$$SE = \sqrt{\frac{0.75 \times 0.25}{2500}}$$

Question 1

Suppose you are taking a multiple-choice test with 4 choices per question. If you use a random number table to guess answers, what is the theoretical probability of guessing a question correctly?

Accepted Answer

$$\frac{1}{4}$$

Question 2

Given the following two-way table summarizing a student body:

Male: Undergraduate 1532, Graduate 458
Female: Undergraduate 1485, Graduate 512

What is $$P(	ext{Male and Graduate})$$?

Accepted Answer

$$\frac{458}{3987}$$

Question 3

When rolling two fair six-sided dice, what is the probability that the sum is 5?

Accepted Answer

$$\frac{4}{36}$$

Question 4

If you roll two dice, what is the probability that the sum is 5 or the sum is 6?

Accepted Answer

$$\frac{9}{36}$$

Question 5

A baby hippo has a mean weight of 88 pounds and a standard deviation of 10 pounds. What is the z-score for a baby hippo weighing 110 pounds?

Accepted Answer

$$z = \frac{110 - 88}{10} = 2.2$$

Question 6

You randomly sample 400 students and find that 72% support a new policy. What is the 95% confidence interval for the proportion of students who support the policy? (Use $$z^* = 1.96)$$

Accepted Answer

$$0.72 \pm 1.96 	imes \sqrt{\frac{0.72 	imes 0.28}{400}}$$

Question 7

If you want to increase the precision of your confidence interval estimate, what should you do?

Accepted Answer

Increase the sample size

Question 8

A company finds that 62% of 234 Texas consumers are satisfied with their product. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{145}{234}$$

Question 9

If you roll two dice, what is the probability that one die shows a 3 and the sum is 7?

Accepted Answer

$$\frac{2}{36}$$

Question 10

Which of the following events are mutually exclusive when rolling two dice?

Accepted Answer

Sum is 5 and sum is 6

Question 11

If you simulate rolling two dice 36 times, what is the expected empirical probability of rolling a sum of 12?

Accepted Answer

$$\frac{1}{36}$$

Question 12

A normal distribution has mean $$\mu = 88$$ and standard deviation $$\sigma = 10. $$What is the probability that a randomly selected baby hippo weighs less than 40 pounds?

Accepted Answer

Very close to 0

Question 13

You calculate a 95% confidence interval for a proportion as (0.14, 0.22). What does this mean in the context of a survey?

Accepted Answer

We are 95% confident that the true proportion is between 0.14 and 0.22

Question 14

If you calculate a 90% confidence interval instead of a 95% confidence interval, what happens to the width of the interval?

Accepted Answer

It becomes narrower

Question 15

If you take 100 samples of size 400 and calculate 95% confidence intervals for each, about how many intervals would you expect to capture the true population proportion?

Accepted Answer

About 95

Question 16

A company collects results from 300 consumers in Kansas and finds 75% are satisfied. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{225}{300}$$

Question 17

In a probability distribution for a dice game, you win 10 points if the sum is less than 4, 3 points if the sum is 4, 5, or 6, and lose 2 points if the sum is 7. What is the expected value of your winnings per roll?

Accepted Answer

$$E = 10 	imes \frac{3}{36} + 3 	imes \frac{9}{36} - 2 	imes \frac{6}{36}$$

Question 18

A survey of 2500 students finds that 75% plan to attend an event. What is the standard error for the proportion?

Accepted Answer

$$SE = \sqrt{\frac{0.75 	imes 0.25}{2500}}$$

Question 19

In a normal distribution, what is the probability that a z-score will be greater than 1.5?

Accepted Answer

About 0.067

Probability, Distributions, and Confidence Intervals: Study Notes for Statistics

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