Q: A sample of 50 batteries has a mean lifetime of 200 hours and a standard deviation of 20 hours. Construct a 95% confidence interval for the population mean battery lifetime. (Use $$z^* = 1.96.)$$

$$200 \pm 1.96 \times \frac{20}{\sqrt{50}}$$

Question 1

Which of the following is a key difference between a binomial and a hypergeometric random variable?

Accepted Answer

A binomial random variable models the number of successes in independent trials with replacement, while a hypergeometric random variable models the number of successes in dependent trials without replacement.

Question 2

Suppose a call center receives an average of 4 calls per hour. What is the probability that exactly 6 calls are received in a given hour? (Use the Poisson distribution.)

Accepted Answer

$$P(X=6) = \frac{4^6$ e$$^{-4}$$}{6!}$$

Question 3

A normal random variable has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the $$z-$$score for an observed value $$x = 130$$?

Accepted Answer

$$z = \frac{130 - 100}{15} = 2$$

Question 4

Which of the following statements about the Central Limit Theorem (CLT) is correct?

Accepted Answer

For a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normal, regardless of the population's distribution.

Question 5

A sample of 50 batteries has a mean lifetime of 200 hours and a standard deviation of 20 hours. Construct a 95% confidence interval for the population mean battery lifetime. (Use $$z^* = 1.96.)$$

Accepted Answer

$$200 \pm 1.96 	imes \frac{20}{\sqrt{50}}$$

Question 6

Which of the following is true about the margin of error in a confidence interval for the mean?

Accepted Answer

It is equal to the critical value times the standard error of the mean.

Question 7

A researcher wants to estimate the proportion of defective products in a large shipment. She samples 100 products and finds 8 are defective. What is the point estimate for the population proportion?

Accepted Answer

$$\hat{p} = \frac{8}{100} = 0.08$$

Question 8

Which of the following best describes a Type I error in hypothesis testing?

Accepted Answer

Rejecting a true null hypothesis.

Question 9

A company claims that at least 90% of its deliveries arrive on time. You wish to test this claim at the 5% significance level. What are the correct null and alternative hypotheses?

Accepted Answer

$$H_0$$: p = 0.90$$; H_a$$: p \u003c 0.90$$

Question 10

If the p-value for a hypothesis test is 0.03 and the significance level $$\alpha is 0.05$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis; there is sufficient evidence to support the alternative hypothesis.

Question 11

Which of the following is NOT a requirement for using the normal approximation for the sampling distribution of the sample proportion?

Accepted Answer

The population must be normally distributed.

Question 12

A small sample ($$n=12) is $$drawn from a population that is approximately normal. Which distribution should be used to construct a confidence interval for the mean?

Accepted Answer

The $$t$$ distribution with $$df = 11$$

Question 13

Which of the following statements about unbiased estimators is correct?

Accepted Answer

An estimator is unbiased if its expected value equals the population parameter.

Question 14

A sample mean is calculated as $$\bar{x} = 50$$ from a sample of $$n = 36$$ with a known population standard deviation $$\sigma = 12. $$What is the standard error of the mean?

Accepted Answer

$$\frac{12}{\sqrt{36}} = 2$$

Question 15

Which of the following best describes the role of the significance level $$\alpha in $$hypothesis testing?

Accepted Answer

It is the probability of making a Type I error.

Question 16

A two-tailed hypothesis test is being conducted at the $$\alpha = 0.01$$ significance level. What is the area in each tail of the distribution?

Accepted Answer

$$0.005$$

Question 17

Which of the following is true about the $$t$$ distribution as the sample size increases?

Accepted Answer

It approaches the standard normal ($$z) $$distribution.

Question 18

A quality control manager wants to determine if the mean weight of cereal boxes is different from the advertised weight of 500g. Which of the following is the correct alternative hypothesis?

Accepted Answer

$$H_a: \mu 
eq 500$$

Question 19

If the sample size increases while all else remains constant, what happens to the width of a confidence interval for the mean?

Accepted Answer

It decreases.

Random Variables, Sampling Distributions, Confidence Intervals, and Hypothesis Testing: Study Guide

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