Question 1

Suppose a population has mean $$\mu = 150$$ and standard deviation $$\sigma = 60. If $$Peter takes a random sample of $$n = 32$$ shifts, what is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\sigma / \sqrt{n} = 60 / \sqrt{32} \approx 10.6$$

Question 2

Which statement best describes the Central Limit Theorem (CLT)?

Accepted Answer

As the sample size $$n$$ increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the population's shape.

Question 3

If a population is not normally distributed, what happens to the shape of the sampling distribution of the sample mean as $$n$$ increases?

Accepted Answer

It becomes more bell-shaped and approaches normality.

Question 4

A researcher collects a random sample of size $$n$$ from a population with mean $$\mu$$ and standard deviation $$\sigma. $$What is the mean of the sampling distribution of the sample mean?

Accepted Answer

$$\mu$$

Question 5

Why does increasing the sample size $$n$$ reduce the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

Because $$\sigma / \sqrt{n}$$ decreases as $$n$$ increases.

Question 6

Peter wants to estimate his average salary per shift. If he increases the number of pay stubs he samples from $$n = 32 to$$ $$n = 64$$, what happens to the standard deviation of the sampling distribution?

Accepted Answer

It decreases by a factor of $$\sqrt{2}.$$

Question 7

If the population distribution is normal, what is the shape of the sampling distribution of the sample mean for any sample size $$n$$?

Accepted Answer

It is always normal.

Question 8

A population has $$\mu = 100$$ and $$\sigma = 20. $$What is the probability that the sample mean from a large sample falls within 2 standard deviations of the population mean?

Accepted Answer

Approximately 0.95

Question 9

Which formula gives the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\sigma / \sqrt{n}$$

Question 10

A population is highly skewed. What can be said about the sampling distribution of the sample mean for $$n = 100$$?

Accepted Answer

It will be approximately normal due to the Central Limit Theorem.

Question 11

If Peter's mean salary per shift is computed from a sample of $$n = 32$$ pay stubs, and the population mean is $$\mu = 150$$, what value would you expect his sample mean to fluctuate around?

Accepted Answer

$$150$$

Question 12

Why is the Central Limit Theorem important in practical statistics?

Accepted Answer

It allows us to make inferences about population means even when the population distribution is unknown or irregular.

Question 13

If the standard deviation of the population is $$\sigma = 20$$ and the sample size is $$n = 25$$, what is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$20 / \sqrt{25} = 4$$

Question 14

A population has mean $$\mu$$ and standard deviation $$\sigma. If $$you take many samples of size $$n$$, what will the distribution of the sample means look like as $$n$$ becomes very large?

Accepted Answer

It will become more normal, regardless of the population's shape.

Question 15

If the sample size $$n is $$small and the population is not normal, what can be said about the sampling distribution of the sample mean?

Accepted Answer

It may not be normal.

Question 16

Which of the following is NOT a consequence of the Central Limit Theorem?

Accepted Answer

The sample mean is always equal to the population mean.

Question 17

A population has $$\mu = 200$$ and $$\sigma = 50. If $$you take a sample of $$n = 100$$, what is the expected mean and standard deviation of the sampling distribution of the sample mean?

Accepted Answer

Mean: $$200$$, SD: $$50 / \sqrt{100} = 5$$

Question 18

If the population mean is $$\mu$$ and the standard deviation is $$\sigma$$, what is the formula for the standard deviation of the sample mean for a sample of size $$n$$?

Accepted Answer

$$\sigma / \sqrt{n}$$

Question 19

In a real-world scenario, why is it beneficial to use a large sample size when estimating a population mean?

Accepted Answer

Because the sample mean will be closer to the population mean and the variability will be lower.

(Lecture 17) Sampling Distributions and the Central Limit Theorem

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