Q: A life insurance policy pays $$100,000 at death or 50$$,000 for permanent disability, charging $500 annually. If the probability of death is 0.001, the probability of permanent disability is 0.004, and the probability of neither is 0.995, what is the expected payout per policyholder?

$$100,000 \times 0.001 + 50$$,000 \times 0.004 + $$0 \times 0.995 = 300$$ + $$200 = 500$$

Q: Suppose you are testing $$H_0$$: p = 0.2$$ against H_a$$: p \neq 0.2$$. You observe $$\hat{p} = 0.25$$ in a sample of n = 100$. What is the standard deviation of the sampling distribution of $$\hat{p}$$?

$$\sqrt{\frac{0.2 \times 0.8}{100}} = 0.04$$

Question 1

A life insurance policy pays $$100,000 at death or 50$$,000 for permanent disability, charging $500 annually. If the probability of death is 0.001, the probability of permanent disability is 0.004, and the probability of neither is 0.995, what is the expected payout per policyholder?

Accepted Answer

$$100,000 	imes 0.001 + 50$$,000 	imes 0.004 + $$0 	imes 0.995 = 300$$ + $$200 = 500$$

Question 2

Which of the following is NOT a requirement for a random variable to be considered a Bernoulli trial?

Accepted Answer

The trials are dependent on each other.

Question 3

Suppose $$X is a $$discrete random variable with possible outcomes 1, 2, 3, 4, and 5, each equally likely. What is the probability that $$X = 3$$?

Accepted Answer

$$\frac{1}{5}$$

Question 4

A company surveys 20 recent graduates to estimate the mean starting salary. Which of the following is the most significant concern when reporting a confidence interval based on this sample?

Accepted Answer

The sample size is too small to ensure the Central Limit Theorem applies.

Question 5

If a sample of size $$n = 60 is $$used instead of $$n = 20 to $$estimate a mean, what is the effect on the confidence interval for the mean?

Accepted Answer

The confidence interval becomes more precise (narrower).

Question 6

Which of the following best describes the Central Limit Theorem?

Accepted Answer

The sampling distribution of the mean becomes Normal as the sample size increases, regardless of the population distribution.

Question 7

Given a normal distribution with $$\mu = 9$$ and $$\sigma = 1.5$$, what is the $$z-$$score for $$x = 12$$?

Accepted Answer

$$z = \frac{12 - 9}{1.5} = 2$$

Question 8

If $$P(Z \u003e 2.0) = 0.0228$$ for a standard normal distribution, what is $$P(Z \leq 2.0)$$?

Accepted Answer

$$0.9772$$

Question 9

A sample proportion $$\hat{p} is $$calculated from a survey. Which of the following conditions must be met for the sampling distribution of $$\hat{p} to be $$approximately Normal?

Accepted Answer

Both $$np \geq 10$$ and $$nq \geq 10$$

Question 10

Which of the following is the correct formula for the variance of the sum of two independent random variables $$P_1$$ and $$P_2$$?

Accepted Answer

$$\mathrm{Var}(P_1$ + P_2$) = \mathrm{Var}(P_1$) + \mathrm{Var}(P_2$)$$

Question 11

A company wants to estimate the proportion of customers who prefer a new product. Which of the following is NOT a required condition for the sampling distribution of the sample proportion to be approximately Normal?

Accepted Answer

The population must be Normally distributed.

Question 12

Which of the following statements about the p-value in hypothesis testing is correct?

Accepted Answer

The p-value is the probability of observing a result as extreme or more extreme than the observed, assuming the null hypothesis is true.

Question 13

Suppose you are testing $$H_0$$: p = 0.2$$ against H_a$$: p 
eq 0.2$$. You observe $$\hat{p} = 0.25$$ in a sample of n = 100$. What is the standard deviation of the sampling distribution of $$\hat{p}$$?

Accepted Answer

$$\sqrt{\frac{0.2 	imes 0.8}{100}} = 0.04$$

Question 14

A researcher collects a simple random sample of 200 people from a city of 5,000. Which sampling condition is being checked by ensuring $$n \leq 0.1N$$?

Accepted Answer

10% Condition

Question 15

If $$X is a $$continuous random variable, which of the following statements is true?

Accepted Answer

The probability that $$X$$ takes any exact value is zero.

Question 16

A basketball player has a free throw success rate of 80%. If she takes 5 shots, what is the probability that she makes exactly 4 shots? (Use the binomial formula: $$P(X = k) = {n \choose k} p^k (1-p$$)^{n-k}$$)

Accepted Answer

$$5 \choose 4 	imes 0.8^4$ 	imes 0.2^1$ = 5 	imes 0.4096 	imes 0.2 = 0.4096$$

Question 17

Which of the following is the correct formula for the $$z-$$score of a sample proportion $$\hat{p}$$ when testing $$H_0$$: p = $$p_0$$?

Accepted Answer

$$z = \frac{\hat{p} - p_0$}{\sqrt{\frac{p_0$ q_0$}{n}}}$$

Question 18

A pharmaceutical company tests a new drug and finds a sample mean effect of 12 units with a sample size of 60. The population standard deviation is unknown. Which model should be used to construct a confidence interval for the mean?

Accepted Answer

t-model

Question 19

If $$X$$ and $$Y$$ are independent random variables with $$E(X) = 5$$, $$E(Y) = 7$$, what is $$E(X + Y)$$?

Accepted Answer

$$E(X + Y) = 12$$

Random Variables, Probability Models, Sampling Distributions, and Hypothesis Testing

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