Q: A fair die is rolled 120 times. What is the mean and standard deviation for the number of times a '6' is rolled, using the normal approximation?

Mean: $$\mu = 20$$, SD: $$\sigma = \sqrt{120 \times \frac{1}{6} \times \frac{5}{6}}$$

Q: A border agency estimates 60% of cars contain undeclared goods. If 70 cars are sampled, what is the mean and standard deviation for the number of cars with undeclared goods?

Mean: $$\mu = 42$$, SD: $$\sigma = \sqrt{70 \times 0.6 \times 0.4}$$

Question 1

Which condition must be satisfied for the normal approximation to the binomial distribution to be considered accurate?

Accepted Answer

$$np \geq 10$$ and $$n(1-p) \geq 10$$

Question 2

When using the normal approximation for a binomial probability, what is the purpose of the continuity correction?

Accepted Answer

To adjust for the difference between discrete and continuous distributions by shifting the interval by 0.5

Question 3

A fair die is rolled 120 times. What is the mean and standard deviation for the number of times a '6' is rolled, using the normal approximation?

Accepted Answer

Mean: $$\mu = 20$$, SD: $$\sigma = \sqrt{120 	imes \frac{1}{6} 	imes \frac{5}{6}}$$

Question 4

In the context of sampling distributions, what does the term 'sampling error' refer to?

Accepted Answer

The difference between the sample statistic and the population parameter

Question 5

If Tim Hortons claims 1 out of 6 cups is a winner, and in a sample of 120 cups, 12 are winners, what is the sampling error?

Accepted Answer

$$0.1 - \frac{1}{6}$$

Question 6

What is the mean (expected value) of the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

$$\mu_{\hat{p}} = p$$

Question 7

What is the standard deviation (standard error) of the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

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

Question 8

Which of the following best describes a sampling distribution?

Accepted Answer

The distribution of all possible values of a sample statistic from repeated samples

Question 9

A border agency estimates 60% of cars contain undeclared goods. If 70 cars are sampled, what is the mean and standard deviation for the number of cars with undeclared goods?

Accepted Answer

Mean: $$\mu = 42$$, SD: $$\sigma = \sqrt{70 	imes 0.6 	imes 0.4}$$

Question 10

Why does the Central Limit Theorem (CLT) allow us to use the normal distribution to approximate the sampling distribution of $$\hat{p}$$?

Accepted Answer

Because as sample size increases, the distribution of $$\hat{p}$$ becomes approximately normal

Question 11

In the die-rolling example, if you want to approximate $$P(k \geq 8)$$ using the normal distribution, what value should you use for the lower bound after applying the continuity correction?

Accepted Answer

$$7.5$$

Question 12

Which of the following is NOT a valid reason for using sample statistics to estimate population parameters?

Accepted Answer

Sample statistics are always equal to population parameters

Question 13

If you consistently observe 30 or more '6's in 120 die rolls, what statistical conclusion might you draw?

Accepted Answer

The die may not be fair, since the observed frequency is much higher than expected

Question 14

Which sampling method involves dividing the population into groups and randomly selecting entire groups?

Accepted Answer

Cluster sampling

Question 15

If the population proportion is $$p = 0.2$$ and the sample size is $$n = 100$$, what is the standard error of $$\hat{p}$$?

Accepted Answer

$$\sqrt{\frac{0.2 	imes 0.8}{100}}$$

Question 16

Which of the following statements about the sampling distribution of $$\hat{p} is $$TRUE?

Accepted Answer

The mean of the sampling distribution equals the population proportion

Question 17

What is the main reason for quantifying sampling error in statistical studies?

Accepted Answer

To understand how far sample estimates may be from the true population value

Question 18

Which equation represents the standard deviation of the binomial distribution used in the normal approximation?

Accepted Answer

$$\sigma = \sqrt{np(1-p)}$$

Question 19

In a study, 60% of cars are estimated to contain undeclared goods. If you sample 70 cars, which calculation would you use to check if the normal approximation is valid?

Accepted Answer

Check if $$70 	imes 0.6 \geq 10$$ and $$70 	imes 0.4 \geq 10$$

Normal Approximation to the Binomial and Sampling Distributions

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