Question 1

Which of the following best describes a discrete random variable?

Accepted Answer

A variable that takes on only a countable set of values, typically associated with counting.

Question 2

A binomial experiment is characterized by which of the following conditions?

Accepted Answer

The experiment has a fixed number of trials, each trial is independent, and there are only two possible outcomes.

Question 3

Which formula correctly represents the variance of a discrete random variable $$X$$ with possible values $$x_i$$ and probabilities $$p(x_i)$$?

Accepted Answer

$$\sum (x_i - \mu)^2$ \cdot p(x_i)$$

Question 4

In a Poisson experiment, what does the parameter $$\lambda$$ represent?

Accepted Answer

The historic average number of events per unit of space or time.

Question 5

A company wants to estimate the average time employees spend on a task. If the sample size is large ($$n \geq 30$$), which distribution should be used to construct a confidence interval for the mean?

Accepted Answer

Normal distribution

Question 6

Which statement about the Central Limit Theorem is correct?

Accepted Answer

It states that for large sample sizes, the sampling distribution of the sample mean will be approximately normal regardless of the population distribution.

Question 7

If a sample proportion is calculated from a sample of size $$n$$, what is the formula for the standard deviation of its sampling distribution?

Accepted Answer

$$\sqrt{\frac{p(1-p)}{n}}$$

Question 8

A confidence interval for the population mean $$\mu is $$constructed as $$\bar{x} \pm z^* \frac{s}{\sqrt{n}}. $$What does $$z^*$$ represent?

Accepted Answer

The critical value from the standard normal distribution corresponding to the desired confidence level

Question 9

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

Accepted Answer

Rejecting a null hypothesis that is true.

Question 10

A researcher is testing whether a new drug improves recovery time. The null hypothesis is $$H_0$$: \mu = 10$$ days. The alternative hypothesis is H_a$$: \mu \u003c 10$$ days. What type of test is this?

Accepted Answer

One-tailed lower test

Question 11

If the p-value in a hypothesis test is less than the significance level $$\alpha$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis and accept the alternative hypothesis.

Question 12

Which formula is used to calculate the margin of error for a confidence interval for a population proportion $$p$$?

Accepted Answer

$$z^* \sqrt{\frac{p(1-p)}{n}}$$

Question 13

Which of the following is true about the t-distribution used in small sample confidence intervals?

Accepted Answer

It becomes narrower as the sample size increases, reflecting reduced uncertainty.

Question 14

A sample of 50 students has a mean test score of 80 and a standard deviation of 10. Construct a 95% confidence interval for the population mean. Which formula should be used?

Accepted Answer

$$\bar{x} \pm z^* \frac{s}{\sqrt{n}}$$

Question 15

Which of the following is an unbiased estimator for the population mean?

Accepted Answer

Sample mean

Question 16

In a hypothesis test for a population proportion $$p$$, which condition must be met for the sampling distribution of $$\hat{p} to be $$approximately normal?

Accepted Answer

There must be at least 15 successes and 15 failures in the sample.

Question 17

A hypergeometric random variable is most appropriate for which scenario?

Accepted Answer

Finding the probability of a certain composition in a sample drawn without replacement from a known population.

Question 18

If the sample size increases, what happens to the standard error of the sample mean?

Accepted Answer

It decreases.

Question 19

Which of the following statements about the rejection region in hypothesis testing is correct?

Accepted Answer

The rejection region is defined by the critical value(s) and occurs in one or both tails of the distribution.

Question 20

A business wants to ensure its estimate of the proportion of customers satisfied with service has a margin of error no greater than 0.03 at 95% confidence. If no prior estimate of $$p is $$available, what value should be used for $$p in $$the sample size formula?

Accepted Answer

0.5

Random Variables, Sampling Distributions, Confidence Intervals, and Hypothesis Testing: Key Concepts for Business Statistics

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