Question 1

Which of the following is NOT a requirement for a probability distribution of a discrete random variable $$X$$?

Accepted Answer

The mean of the distribution must be 0.

Question 2

A pharmaceutical company tests a new drug on 100 patients and records whether each patient recovers (success) or not (failure). Which of the following best describes this experiment?

Accepted Answer

A binomial experiment, because there are a fixed number of independent trials with two possible outcomes.

Question 3

Given a binomial distribution with $$n=20$$ and $$p=0.3$$, what is the variance of the distribution?

Accepted Answer

$$4.2$$

Question 4

Find the area under the standard normal curve to the left of $$z = 1.25.$$

Accepted Answer

$$0.8944$$

Question 5

A random variable $$X is $$normally distributed with mean $$\mu = 50$$ and standard deviation $$\sigma = 8. $$What is the probability that $$X is $$less than $$58$$?

Accepted Answer

$$0.8413$$

Question 6

Find the $$z-$$score that corresponds to the $$90th $$percentile.

Accepted Answer

$$z = 1.28$$

Question 7

A student scores $$85 on a $$test with a mean of $$80$$ and a standard deviation of $$5. $$Another student scores $$78 on a $$different test with a mean of $$70$$ and a standard deviation of $$4. $$Who performed better relative to their class?

Accepted Answer

The second student, because their $$z-$$score is higher.

Question 8

A sample of $$n=36 is $$taken from a population with mean $$\mu=100$$ and standard deviation $$\sigma=12. $$What is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$2$$

Question 9

Given a confidence level of $$95\%$$, a population standard deviation $$\sigma = 10$$, and a sample size $$n = 25$$, what is the margin of error for the confidence interval for the mean?

Accepted Answer

$$3.92$$

Question 10

A researcher wants to estimate the mean cholesterol level in a population to within $$\pm 3$$ units with $$99\%$$ confidence. If the population standard deviation is $$12$$, what is the minimum sample size required?

Accepted Answer

$$n = 107$$

Question 11

Which of the following scenarios requires the use of a $$t-$$distribution rather than a $$z-$$distribution?

Accepted Answer

The population standard deviation is unknown and the sample size is small.

Question 12

Suppose you conduct a hypothesis test with $$\alpha = 0.05$$ and obtain a $$p-$$value of $$0.03. $$What is the correct conclusion?

Accepted Answer

Reject the null hypothesis.

Question 13

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

Accepted Answer

Failing to reject a false null hypothesis.

Question 14

A study compares the effect of two diets on the same group of patients, measuring their weight before and after the diet. What type of samples are these?

Accepted Answer

Dependent samples.

Question 15

Given a left-tailed test with $$\alpha = 0.01$$, what is the critical value for $$z$$?

Accepted Answer

$$-2.33$$

Question 16

A 95\% confidence interval for the mean is $$(12, 18). A $$researcher claims the population mean is $$20. $$What is the correct conclusion?

Accepted Answer

Reject the claim, because $$20 is $$not in the confidence interval.

Question 17

Which of the following is NOT a condition for using a one-sample $$z$$ test for the mean?

Accepted Answer

The population standard deviation is unknown.

Question 18

A hospital administrator wants to test if the average wait time is different from 30 minutes. What type of test should be used?

Accepted Answer

A two-tailed test.

Question 19

If $$P(Z \u003c z) = 0.975$$, what is the value of $$z$$?

Accepted Answer

$$1.96$$

Question 20

A sample of $$n=16$$ has a mean of $$52$$ and a standard deviation of $$8. $$What is the $$t-$$statistic for testing $$H_0$$: \mu = 50$$?

Accepted Answer

$$t = 1.0$$

Midterm 2 Study Guide: Probability, Distributions, and Hypothesis Testing

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