Q: A used car dealer has 76 cars that were produced in 2023, and about 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 mean $$\mu$$ and standard deviation $$\sigma$$ for the number of recalled cars in the dealer's lot?

$$\mu = 76 \times \frac{35}{94}$$, $$\sigma = \sqrt{76 \times \frac{35}{94} \times \left(1 - \frac{35}{94}\right)}$$

Question 1

According to the Central Limit Theorem (CLT), as the sample size $$n$$ increases, what happens to the sampling distribution of the sample mean $$\bar{X}$$?

Accepted Answer

It becomes 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 students at a university have. According to the Central Limit Theorem, what should the researcher do to make their sampling distribution get closer to normal?

Accepted Answer

Increase the sample size.

Question 3

Suppose the mean $$\mu_X = 3.2$$ and the standard deviation $$\sigma_X = 0.98. $$What is the standard deviation of the sampling distribution of the sample mean for samples of size $$n = 60$$?

Accepted Answer

$$\sigma_{\bar{X}} = \frac{0.98}{\sqrt{60}}$$

Question 4

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 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 $$z-$$score formula to find 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

$$z = \frac{4.3 - 4.2}{0.25 / \sqrt{36}}$$

Question 6

When using the normal distribution to approximate a binomial probability, 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 a random sample of 100 people reveals at least 60 people preferring Pepsi. Which continuity correction should be used?

Accepted Answer

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

Question 8

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

Accepted Answer

The actual number of successes divided by the number of trials.

Question 9

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

Accepted Answer

Mean: $$\mu_X = 5$$, Standard deviation: $$\sigma_X = 1.58$$

Question 10

A used car dealer has 76 cars that were produced in 2023, and about 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 mean $$\mu$$ and standard deviation $$\sigma$$ for the number of recalled cars in the dealer's lot?

Accepted Answer

$$\mu = 76 	imes \frac{35}{94}$$, $$\sigma = \sqrt{76 	imes \frac{35}{94} 	imes \left(1 - \frac{35}{94}ight)}$$

Question 11

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

Accepted Answer

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

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 sample proportion $$\hat{p}$$?

Accepted Answer

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

Question 13

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

Accepted Answer

It is the 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 that the sampling distribution is normal?

Accepted Answer

No, because the sample size is small and the population distribution may not be normal.

Question 15

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

Accepted Answer

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

Question 16

When using the normal distribution to approximate a binomial probability, 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 17

A previous study found that 80% of people preferred drinking Pepsi over Coca Cola. Use a normal distribution to approximate the probability that, from a random sample of 100 people, between 10 and 11 people prefer Coca Cola. What is the value of $$p$$ for this calculation?

Accepted Answer

$$p = 0.2$$

Question 18

For a binomial variable $$X$$ with $$n = 10$$ and $$p = 0.5$$, what is the mean and standard deviation of the sample proportion $$\hat{p}$$?

Accepted Answer

Mean: $$\mu_{\hat{p}} = 0.5$$, Standard deviation: $$\sigma_{\hat{p}} = \sqrt{\frac{0.5 	imes 0.5}{10}}$$

Sampling Distributions, Central Limit Theorem, and Confidence Intervals in Statistics

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