Q: A sample proportion $$p is $$approximately normally distributed if which of the following conditions are met?

$$n\pi \geq 5$$ \text{ and } $$n(1-\pi) \geq 5$$

Question 1

Which of the following best describes a sampling distribution?

Accepted Answer

A distribution of all possible values of a sample statistic for a given sample size selected from a population.

Question 2

Given a population with values 18, 20, 22, and 24, what is the mean ($$\mu) of $$the population?

Accepted Answer

$$\mu = \frac{18 + 20 + 22 + 24}{4} = 21$$

Question 3

What happens to the standard error of the mean as the sample size increases?

Accepted Answer

It decreases.

Question 4

If a population is normal with mean $$\mu$$ and standard deviation $$\sigma$$, what is the standard deviation of the sampling distribution of the mean for samples of size $$n$$?

Accepted Answer

$$\sigma_{\overline{X}} = \frac{\sigma}{\sqrt{n}}$$

Question 5

A population has mean $$\mu = 8$$ and standard deviation $$\sigma = 3. $$For a sample of size $$n = 36$$, what is the probability that the sample mean is between 7.8 and 8.2?

Accepted Answer

0.3108

Question 6

Which statement about the Central Limit Theorem is correct?

Accepted Answer

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

Question 7

For most distributions, what is the minimum sample size $$n$$ required for the sampling distribution of the mean to be approximately normal?

Accepted Answer

$$n \u003e 30$$

Question 8

If the true proportion of voters who support Proposition A is $$\pi = 0.4$$, what is the probability that a sample of size $$n = 200$$ yields a sample proportion between 0.40 and 0.45?

Accepted Answer

0.4251

Question 9

Which formula is used to standardize a sample mean to a Z value?

Accepted Answer

$$Z = \frac{\overline{X} - \mu}{\sigma_{\overline{X}}}$$

Question 10

A sample proportion $$p is $$approximately normally distributed if which of the following conditions are met?

Accepted Answer

$$n\pi \geq 5$$ 	ext{ and } $$n(1-\pi) \geq 5$$

Question 11

Which of the following is true about the mean of the sampling distribution of the sample mean?

Accepted Answer

$$\mu_{\overline{X}} = \mu$$

Question 12

Suppose you want to find an interval that includes 95% of the sample means for $$\mu = 368$$, $$\sigma = 15$$, and $$n = 25. $$What are the lower and upper limits?

Accepted Answer

Lower: $$362.12$$, Upper: $$373.88$$

Question 13

Which formula gives the standard error of the sample proportion?

Accepted Answer

$$\sigma_p = \sqrt{\frac{\pi(1-\pi)}{n}}$$

Question 14

Why is the sample mean considered an unbiased estimator of the population mean?

Accepted Answer

Because the expected value of the sample mean equals the population mean.

Question 15

If the population distribution is not normal, under what condition can the sampling distribution of the sample mean be considered approximately normal?

Accepted Answer

When the sample size is large enough (usually $$n \u003e 30).$$

Sampling Distributions: Concepts, Properties, and Applications

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