Q: For a population with mean $$\mu = 368$$, standard deviation $$\sigma = 15$$, and sample size $$n = 25$$, what is the lower limit of the interval that includes 95% of the sample means?

$$368 + (-1.96) \times \frac{15}{\sqrt{25}} = 362.12$$

Q: A sample mean is calculated from a sample of size $$n = 25$$ from a population with $$\mu = 368$$ and $$\sigma = 15. $$What is the upper limit of the interval that includes 95% of the sample means?

$$368 + (1.96) \times \frac{15}{\sqrt{25}} = 373.88$$

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

Suppose a population consists of the ages 18, 20, 22, and 24. What is the mean age of this population?

Accepted Answer

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

Question 3

If you take all possible samples of size 2 from a population of 4 values (18, 20, 22, 24), how many unique samples are there if sampling is with replacement?

Accepted Answer

$$4 	imes 4 = 16$$

Question 4

What is the formula for the standard deviation of the sampling distribution of the sample mean when the population standard deviation is $$\sigma$$ and the sample size is $$n$$?

Accepted Answer

$$\frac{\sigma}{\sqrt{n}}$$

Question 5

According to the Central Limit Theorem, what happens to the sampling distribution of the sample mean as the sample size increases?

Accepted Answer

It approaches a normal distribution, regardless of the population's shape.

Question 6

For a population with mean $$\mu = 368$$, standard deviation $$\sigma = 15$$, and sample size $$n = 25$$, what is the lower limit of the interval that includes 95% of the sample means?

Accepted Answer

$$368 + (-1.96) 	imes \frac{15}{\sqrt{25}} = 362.12$$

Question 7

A restaurant has a mean wait time of $$\mu = 8$$ minutes and $$\sigma = 3$$ minutes. For a sample of $$n = 36$$, what is the standard deviation of the sampling distribution of the mean?

Accepted Answer

$$\frac{3}{\sqrt{36}} = 0.5$$

Question 8

If the population distribution is normal, what is true about the sampling distribution of the mean for any sample size?

Accepted Answer

It is always normally distributed.

Question 9

Which of the following is an unbiased estimator of the population mean $$\mu$$?

Accepted Answer

The sample mean $$\bar{X}$$

Question 10

A company wants to estimate the average salary of its employees. If the population standard deviation is $$\sigma = 5000$$ and the sample size is $$n = 100$$, what is the standard error of the mean?

Accepted Answer

$$\frac{5000}{\sqrt{100}} = 500$$

Question 11

If you want the sampling distribution of the mean to be approximately normal for a skewed population, what is the minimum recommended sample size?

Accepted Answer

$$n \u003e 30$$

Question 12

Given a population with $$\mu = 21$$ and $$\sigma = 2.236$$, what is the $$z-$$value for a sample mean $$\bar{X} = 22$$ from a sample of $$n = 4$$?

Accepted Answer

$$\frac{22 - 21}{2.236 / \sqrt{4}} = \frac{1}{1.118} = 0.894$$

Question 13

Why does the sampling distribution of the sample mean tend to be less variable than the population distribution?

Accepted Answer

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

Question 14

A sample mean is calculated from a sample of size $$n = 25$$ from a population with $$\mu = 368$$ and $$\sigma = 15. $$What is the upper limit of the interval that includes 95% of the sample means?

Accepted Answer

$$368 + (1.96) 	imes \frac{15}{\sqrt{25}} = 373.88$$

Question 15

If 95% of sample means fall within a certain interval, what percentage of sample means fall above the upper limit of this interval?

Accepted Answer

2.5%

Question 16

Which statement about the Central Limit Theorem is correct?

Accepted Answer

It states that the sampling distribution of the sample mean approaches normality as sample size increases, regardless of the population's distribution.

Question 17

A business analyst collects 50 samples of size 50 from a college to estimate mean GPA. What is the main reason for using the sampling distribution of the mean in this context?

Accepted Answer

To understand the variability and distribution of possible sample means for inference about the population mean.

Question 18

If the population mean is $$\mu$$ and the sample mean is $$\bar{X}$$, which formula gives the $$z-$$value for the sampling distribution of the mean?

Accepted Answer

$$z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}}$$

Sampling Distributions – Statistics for Business

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