Q: A company has 100 customers with an average customer service rating of 7.2 and a standard deviation of 0.7. A sample of 40 customers gives a mean rating of 7.5. If sampling is without replacement, what formula should be used to calculate the standard error of the mean?

$$\\sigma_{\\bar{x}} = \\frac{\\sigma}{\\sqrt{n}} \\sqrt{\\frac{N-n}{N-1}}$$

Q: A poll claims that 60% of voters support a candidate ($$p = 0.6). In a $$sample of $$n = 250$$, only 52% support the candidate. What is the formula for the z-score to test this claim?

$$z = \\frac{P - p}{\\sqrt{\\frac{p(1-p)}{n}}}$$

Q: A college claims that 70% of its 770 graduates found jobs related to their majors. In a sample of 120, 97 found such jobs. What is the sample proportion?

$$\\hat{p} = \\frac{97}{120} = 0.808$$

Q: What is the formula for the standard error of the proportion for a finite population?

$$\\sigma_{\\hat{p}} = \\sqrt{\\frac{p(1-p)}{n} \\times \\frac{N-n}{N-1}}$$

Question 1

Which of the following statements best describes the Central Limit Theorem (CLT)?

Accepted Answer

The CLT states that the sample means of large-sized samples will be normally distributed, regardless of the shape of their population distributions.

Question 2

A company has 100 customers with an average customer service rating of 7.2 and a standard deviation of 0.7. A sample of 40 customers gives a mean rating of 7.5. If sampling is without replacement, what formula should be used to calculate the standard error of the mean?

Accepted Answer

$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \sqrt{\frac{N-n}{N-1}}$$

Question 3

Which sampling method is most appropriate when you want to ensure that all subgroups (such as age categories or shifts) are represented in your sample?

Accepted Answer

Stratified sampling

Question 4

In systematic sampling, how is the sampling constant $$k$$ calculated?

Accepted Answer

$$k = \frac{N}{n}$$

Question 5

A poll claims that 60% of voters support a candidate ($$p = 0.6). In a $$sample of $$n = 250$$, only 52% support the candidate. What is the formula for the z-score to test this claim?

Accepted Answer

$$z = \frac{P - p}{\sqrt{\frac{p(1-p)}{n}}}$$

Question 6

Which of the following is a potential disadvantage of systematic sampling?

Accepted Answer

Periodicity in the population may lead to biased samples.

Question 7

What is the main difference between stratified sampling and cluster sampling?

Accepted Answer

Stratified sampling divides the population into homogeneous groups based on a characteristic, while cluster sampling divides the population into heterogeneous groups that are mini-subsets of the population.

Question 8

If the sample size $$n$$ increases, what happens to the standard error of the mean?

Accepted Answer

It decreases.

Question 9

A sample statistic is not likely to be exactly equal to the population parameter. What is the term for the difference between a sample statistic and a population parameter?

Accepted Answer

Sampling error

Question 10

Which of the following is NOT a probability sampling method?

Accepted Answer

Convenience sampling

Question 11

A car owner wants to test if the average commute time is less than 45 minutes, given a population mean of 50 minutes and standard deviation of 14 minutes. What is the first step in applying the Central Limit Theorem to this scenario?

Accepted Answer

Take a random sample of trips and compute the sample mean and standard error of the mean.

Question 12

Which formula is used to calculate the sample proportion?

Accepted Answer

$$\hat{p} = \frac{x}{n}$$

Question 13

When is the finite population correction factor necessary in calculating the standard error?

Accepted Answer

When the sample size is more than 5% of the population and sampling is without replacement.

Question 14

A college claims that 70% of its 770 graduates found jobs related to their majors. In a sample of 120, 97 found such jobs. What is the sample proportion?

Accepted Answer

$$\hat{p} = \frac{97}{120} = 0.808$$

Question 15

Which of the following is a nonsampling error?

Accepted Answer

Ambiguous survey questions

Question 16

If the population follows a normal probability distribution, what can be said about the distribution of sample means regardless of sample size?

Accepted Answer

The sample means will also be normally distributed.

Question 17

What is the formula for the standard error of the proportion for a finite population?

Accepted Answer

$$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n} \times \frac{N-n}{N-1}}$$

Question 18

Which resampling method involves repeatedly drawing samples with replacement to estimate a population parameter?

Accepted Answer

Bootstrap method

Question 19

A sample mean falls outside the interval that includes 95% of the means around the claimed population mean. What does this imply?

Accepted Answer

There is sufficient evidence to reject the claim (null hypothesis).

Chapter 7: Sampling and Sampling Distributions – Study Notes for Business Statistics

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