Q: A company surveys 500 customers and finds 400 are satisfied. To check the success/failure condition for a confidence interval for a proportion, which calculation is correct?

$$np = 500 \times 0.8 = 400$$ and $$n(1-p) = 500 \times 0.2 = 100$$

Question 1

A company claims that 60% of its customers are satisfied with their service. You take a random sample of 100 customers and find that 54 are satisfied. Which of the following is the correct form of a 95% confidence interval for the true proportion of satisfied customers?

Accepted Answer

$$\hat{p} \pm z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$$

Question 2

Which of the following statements about the Law of Large Numbers is correct?

Accepted Answer

As the number of trials increases, the observed proportion of outcomes approaches the theoretical probability.

Question 3

Suppose $$X$$ and $$Y$$ are independent random variables with $$E(X) = 5$$ and $$E(Y) = 3. $$What is $$E(X + Y)$$?

Accepted Answer

$$8$$

Question 4

A histogram of exam scores is unimodal, symmetric, and bell-shaped. Which distribution is most appropriate to model these scores?

Accepted Answer

Normal distribution

Question 5

If a random variable $$X is $$transformed by $$Y = 2X + 3$$, how does the standard deviation of $$Y$$ compare to that of $$X$$?

Accepted Answer

The standard deviation of $$Y is$$ $$2$$ times the standard deviation of $$X.$$

Question 6

A survey is conducted to estimate the average time students spend studying per week. The sample mean is $$12$$ hours, the sample standard deviation is $$4$$ hours, and the sample size is $$25. $$What is the standard error of the mean?

Accepted Answer

$$0.8$$

Question 7

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

Accepted Answer

Rejecting a true null hypothesis

Question 8

A manufacturer wants to test if the mean weight of their product is different from $$500$$ grams. Which is the correct null hypothesis?

Accepted Answer

$$H_0$$: \mu = 500$$

Question 9

If the standard deviation of a population is unknown, which distribution should be used to construct a confidence interval for the mean?

Accepted Answer

t-distribution

Question 10

A random sample of 200 voters shows that 120 support a new policy. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$0.6$$

Question 11

Which of the following is NOT a required condition for constructing a confidence interval for a proportion?

Accepted Answer

Population must be normally distributed

Question 12

If $$X$$ and $$Y$$ are independent normal random variables, which of the following statements is true about $$X + Y$$?

Accepted Answer

$$X + Y is $$also normally distributed

Question 13

A normal distribution has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the z-score for a value of $$130$$?

Accepted Answer

$$2$$

Question 14

Which of the following best describes the margin of error in a confidence interval for a proportion?

Accepted Answer

Critical value multiplied by the standard error

Question 15

A business analyst wants to estimate the mean sales per day. She collects a random sample and constructs a 95% confidence interval. Which of the following is the correct interpretation?

Accepted Answer

We are 95% confident the true mean sales per day is in this interval.

Question 16

If the probability of an event is $$0.3$$, what is the probability that the event does NOT occur?

Accepted Answer

$$0.7$$

Question 17

A company surveys 500 customers and finds 400 are satisfied. To check the success/failure condition for a confidence interval for a proportion, which calculation is correct?

Accepted Answer

$$np = 500 	imes 0.8 = 400$$ and $$n(1-p) = 500 	imes 0.2 = 100$$

Question 18

Which of the following is a potential violation when constructing a sampling distribution?

Accepted Answer

Lack of independence among observations

Question 19

A normal distribution has mean $$\mu = 50$$ and standard deviation $$\sigma = 5. $$According to the 68–95–99.7 rule, approximately what percentage of values fall between $$40$$ and $$60$$?

Accepted Answer

About 95%

Question 20

A researcher is testing $$H_0$$: \mu = 100$$ against H_a$$: \mu 
eq 100$$. She finds a p-value of 0.03$ and uses $$\alpha = 0.05$$. What is the correct conclusion?

Accepted Answer

Reject the null hypothesis; there is evidence that $$\mu 
eq 100.$$

Business Statistics: Random Variables, Probability Models, Normal Distributions, and Inference

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