Q: Which formula correctly calculates the sample proportion $$\hat{p}$$?

$$\hat{p} = \frac{\text{Count}}{n}$$

Question 1

Which of the following best describes the difference between a statistic and a parameter?

Accepted Answer

A statistic is a numerical value calculated from a sample, while a parameter is a numerical value calculated from the entire population.

Question 2

Given a population with mean $$\mu = 4.3$$ and standard deviation $$\sigma = 2.1$$, what is the standard deviation of the sampling distribution of the sample mean for samples of size $$n = 9$$?

Accepted Answer

$$\frac{2.1}{\sqrt{9}} = 0.7$$

Question 3

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

Accepted Answer

For large sample sizes ($$n \geq 30$$), the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution.

Question 4

Suppose you take many random samples of size $$n = 5$$ from a population with mean $$\mu = 4.3. $$What would you expect the mean of the sampling distribution of $$\overline{Y} to be$$?

Accepted Answer

$$4.3$$

Question 5

Refer to the first histogram (Population of Word Lengths). Which of the following best describes its shape?

Accepted Answer

Skewed to the right

Question 6

Refer to the second histogram (Distribution of the Y-bars). Compared to the population distribution, the sampling distribution of the sample mean is:

Accepted Answer

Less variable and less skewed

Question 7

If the population standard deviation is $$\sigma = 2.1$$ and you take a sample of size $$n = 49$$, what is the standard error of the sample mean?

Accepted Answer

$$\frac{2.1}{\sqrt{49}} = 0.3$$

Question 8

A researcher wants to estimate the average length of words in a novel. She takes a random sample of 36 words and finds a sample mean of 4.5 letters. If the population standard deviation is known to be 2.1, what is the $$z-$$score for this sample mean, given $$\mu = 4.3$$?

Accepted Answer

$$z = \frac{4.5 - 4.3}{2.1/\sqrt{36}} = \frac{0.2}{0.35} \approx 0.57$$

Question 9

Which of the following is NOT a property of the sampling distribution of the sample mean?

Accepted Answer

It always has the same shape as the population distribution, regardless of sample size.

Question 10

If a population is highly skewed, what can you say about the sampling distribution of the sample mean for samples of size $$n = 40$$?

Accepted Answer

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

Question 11

A sample of 25 patients has a mean cholesterol level of 200 mg/dL. The population standard deviation is 30 mg/dL. What is the probability that a random sample of 25 patients will have a mean cholesterol level greater than 210 mg/dL, assuming the population mean is 200 mg/dL?

Accepted Answer

First, calculate $$z = \frac{210 - 200}{30/\sqrt{25}} = \frac{10}{6} = 1.67. $$The probability is $$P(Z \u003e 1.67) \approx 0.0475.$$

Question 12

Which formula correctly calculates the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{	ext{Count}}{n}$$

Question 13

Why does the sampling distribution of the sample mean have less variability than the population distribution?

Accepted Answer

Because the standard deviation of the sample mean is $$\frac{\sigma}{\sqrt{n}}$$, which is less than $$\sigma$$ for $$n \u003e 1.$$

Question 14

A population has a mean $$\mu = 50$$ and standard deviation $$\sigma = 10. If $$you take a random sample of $$n = 100$$, what is the probability that the sample mean is less than 48?

Accepted Answer

First, calculate $$z = \frac{48 - 50}{10/\sqrt{100}} = \frac{-2}{1} = -2. $$The probability is $$P(Z \u003c -2) \approx 0.0228.$$

Question 15

If you increase the sample size $$n$$, what happens to the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

It decreases, because $$\frac{\sigma}{\sqrt{n}}$$ gets smaller as $$n$$ increases.

Question 16

Which of the following is a correct interpretation of the Central Limit Theorem in the context of medical research?

Accepted Answer

If a researcher takes many large random samples of patient blood pressure, the distribution of the sample means will be approximately normal, even if the original blood pressure data is skewed.

Sampling Distributions and the Sample Mean

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