Question 1

Which of the following best describes a parameter in statistics?

Accepted Answer

A number that reflects properties of a population and is typically an unknown constant.

Question 2

Suppose you take repeated random samples of size $$n$$ from a population with mean $$\mu$$ and standard deviation $$\sigma. $$What is the expected value (mean) of the sampling distribution of the sample mean $$\overline{X}$$?

Accepted Answer

$$\mu$$

Question 3

A researcher wants to estimate the average number of people living in a household in the US. The population mean is $$\mu = 2.9$$ and the standard deviation is $$\sigma = 1.48. If $$she takes a simple random sample of $$n = 16$$ households, what is the standard error of the sample mean?

Accepted Answer

$$1.48 / \sqrt{16} = 0.37$$

Question 4

Which of the following statements about the Central Limit Theorem (CLT) is correct?

Accepted Answer

For large $$n$$, the sampling distribution of the sample mean is approximately normal, regardless of the population's distribution.

Question 5

A poll reports that 41% of the public believes young adults are having the toughest time in today's economy, with a margin of error of $$\pm 2.9\% at 95$$% confidence. What is the correct interpretation of this margin of error?

Accepted Answer

We are 95% confident that the true proportion is between 38.1% and 43.9%.

Question 6

If the standard deviation of a population is $$\sigma = 0.02$$ grams and you take samples of size $$n = 25$$, what is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$0.02 / \sqrt{25} = 0.004$$

Question 7

Which of the following is NOT a step in constructing the sampling distribution of a statistic for small $$N$$ and $$n$$?

Accepted Answer

Calculate the population mean from the sample data.

Question 8

A population is highly skewed to the right. According to the Central Limit Theorem, what happens to the shape of the sampling distribution of the sample mean as $$n$$ increases?

Accepted Answer

It becomes approximately normal as $$n$$ becomes large.

Question 9

Suppose you have a population with mean $$\mu = 18$$ and standard deviation $$\sigma = 5. If $$you take a random sample of size $$n = 45$$, what is the probability that the sample mean $$\overline{X} is $$between 18 and 18.5? (Use the standard normal distribution.)

Accepted Answer

$$0.2486$$

Question 10

Which of the following best describes the standard error (SE) of the sample mean?

Accepted Answer

The standard deviation of the sampling distribution of the sample mean.

Question 11

A basketball coach takes five samples of size $$n = 6$$ and five samples of size $$n = 100$$ from the same population. Which set of sample means will show less variability and why?

Accepted Answer

The $$n = 100$$ samples, because larger samples have less sampling variability.

Question 12

Given a normal population with $$\mu = 105$$ and $$\sigma = 16$$, and a sample size $$n = 20$$, what is the probability that the sample mean $$\overline{X} is $$between 103.2 and 106.8?

Accepted Answer

$$0.383$$

Question 13

Which of the following is true about the mean of the sampling distribution of the sample mean $$\overline{X}$$?

Accepted Answer

It is equal to the population mean.

Question 14

If $$X is a $$random variable with mean $$\mu$$ and standard deviation $$\sigma$$, and $$\overline{X} is $$the sample mean of a sample of size $$n$$, what is the Z-transform for $$\overline{X}$$?

Accepted Answer

$$Z = \frac{\overline{X} - \mu}{\sigma / \sqrt{n}}$$

Question 15

A population is not normal, but you take a sample of size $$n = 40. $$What can you say about the sampling distribution of the sample mean?

Accepted Answer

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

Question 16

Which of the following factors does NOT affect the shape of the sampling distribution of the sample mean?

Accepted Answer

Population mean $$\mu$$

Question 17

If the standard deviation of the sample mean is $$0.0086$$ for $$n = 5$$ and $$0.0045$$ for $$n = 20$$, what does this illustrate about the relationship between sample size and standard error?

Accepted Answer

Standard error decreases as sample size increases.

Question 18

A population has $$\mu = 2.9$$ and $$\sigma = 1.48. $$What is the approximate standard error of the sample mean for $$n = 30$$?

Accepted Answer

$$1.48 / \sqrt{30} \approx 0.27$$

Question 19

Which of the following is a correct statement about unbiased estimators?

Accepted Answer

The sample mean $$\overline{X} is an $$unbiased estimator of the population mean $$\mu.$$

Estimating Parameters and Sampling Distributions: Study Notes xxx

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