Question 1

According to the Central Limit Theorem (CLT), as the sample size $$n$$ increases, the sampling distribution of the sample mean $$\overline{X}$$ becomes:

Accepted Answer

Closer to a normal distribution, regardless of the population distribution

Question 2

A researcher takes 10 samples of 20 students each to get a sampling distribution of the average number of siblings at a university. According to the Central Limit Theorem, what should the researcher do to make the sampling distribution get closer to normal?

Accepted Answer

Increase the sample size $$n in $$each sample

Question 3

A video game retailer collects 100 random samples of 40 players to estimate the average play time of their most popular game. The mean of the sampling distribution is 26.7 hours. In this example, what is the value of $$n$$?

Accepted Answer

$$n = 40$$

Question 4

If $$\mu_X = 3.2$$ and $$\sigma_X = 0.98$$, what is the $$z-$$score for a sample mean $$\overline{X} = 3.6$$ from a sample of $$n = 60$$?

Accepted Answer

$$z = \dfrac{3.6 - 3.2}{0.98 / \sqrt{60}}$$

Question 5

A test audience of 36 moviegoers gives a new film an average score of 4.2 out of 5 stars with a standard deviation of 0.25 stars. What is the probability that a random test audience of 36 moviegoers will give the film an average rating of 4.3 stars or greater?

Accepted Answer

Find $$z = \dfrac{4.3 - 4.2}{0.25 / \sqrt{36}}$$ and use the standard normal table to find $$P(z \geq 	ext{calculated } z)$$

Question 6

When using the normal distribution to approximate binomial probabilities, which of the following conditions must be satisfied?

Accepted Answer

$$np \geq 5$$ and $$nq \geq 5$$

Question 7

A previous study found that 80% of people preferred drinking Pepsi over Coca Cola. Use a normal distribution to approximate the probability that at least 60 people in a random sample of 100 prefer Pepsi. Which continuity correction should be used?

Accepted Answer

Use $$P(X \geq 59.5)$$

Question 8

In a binomial experiment, the sample proportion $$\hat{p} is $$defined as:

Accepted Answer

The actual number of successes divided by the sample size $$n$$

Question 9

For a binomial variable $$X$$ with $$n = 10$$ and $$p = 0.5$$, what is the mean and standard deviation of the sampling distribution of $$X$$?

Accepted Answer

$$\mu_X = np = 5$$, $$\sigma_X = \sqrt{npq} = 1.58$$

Question 10

A used car dealer has 76 cars produced in 2023, and 35 million out of 94 million cars produced that year were recalled. Use a normal distribution to approximate the probability that more than half of these cars were recalled. What is the value of $$p$$ for the binomial distribution?

Accepted Answer

$$p = \dfrac{35}{94}$$

Question 11

To construct a confidence interval for a population proportion, which formula is used for the margin of error $$E$$?

Accepted Answer

$$E = z_{\alpha/2} \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$$

Question 12

A survey of 200 people found that 90 preferred computers from Brand A over Brand B. Construct a 90% confidence interval for the true proportion of people who prefer Brand A. What is the point estimate $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \dfrac{90}{200}$$

Question 13

Which of the following best describes the sampling distribution of the sample mean $$\overline{X}$$?

Accepted Answer

It is the probability distribution of all possible sample means from samples of the same size drawn from the population

Question 14

A company’s marketing team takes 50 samples of 10 recent clients to create a sampling distribution of sample means for the average amount spent per month on company products. Can the Central Limit Theorem be used to determine if the sampling distribution is normal?

Accepted Answer

No, because the sample size $$n = 10$$ may be too small unless the population is normal

Question 15

When approximating a binomial distribution with a normal distribution, what adjustment should be made to the value of $$x$$?

Accepted Answer

Apply a continuity correction by adding or subtracting 0.5 to $$x$$

Question 16

A previous study found that 80% of people preferred drinking Pepsi over Coca Cola. For a sample of 100 people, what is the mean and standard deviation of the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

$$\mu_{\hat{p}} = 0.8$$, $$\sigma_{\hat{p}} = \sqrt{\dfrac{0.8 	imes 0.2}{100}}$$

Question 17

The probability of someone voting for a particular candidate in a two-person election is 45%. Use a normal distribution to approximate the probability that between 62 and 70 people out of a sample of 100 vote for the candidate. Which $$z-$$scores should be calculated?

Accepted Answer

Calculate $$z$$ for $$x = 61.5$$ and $$x = 70.5$$

Question 18

Which of the following is NOT a requirement for using the normal approximation to the binomial distribution?

Accepted Answer

The sample size $$n$$ must be less than 30

Sampling Distributions, Central Limit Theorem, and Confidence Intervals for Proportions

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