## Statistical Techniques in Business and Economics Douglas Lind 17e

*Statistical Techniques in Business and Economics, 17e* (Lind)

**Chapter 8 Sampling Methods and the Central Limit Theorem**

1) Sampling a population is often necessary because the cost of studying all the items in the population is prohibitive.

Answer: TRUE

Explanation: Taking a sample from a population will be far less costly than trying to do a census of the entire population.

Difficulty: 1 Easy

Topic: Sampling Methods

Learning Objective: 08-01 Explain why populations are sampled and describe four methods to sample a population.

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

2) It is often not feasible to study the entire population because it is impossible to observe all the items in the population.

Answer: TRUE

Explanation: If a population is essentially infinite, it is physically impossible to do a census. If you wish to check for water contamination in a lake, you take one or more samples from the lake, but it would not be possible to check all water molecules in the lake.

Difficulty: 1 Easy

Topic: Sampling Methods

Learning Objective: 08-01 Explain why populations are sampled and describe four methods to sample a population.

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

3) When deciding to collect sample information rather than collecting information from the population, the amount of time required to collect the information is unimportant.

Answer: FALSE

Explanation: If a population is large, it may take a very long time to collect the information. To reduce the amount of time to collect the information, a sample of the population may be preferred.

Difficulty: 1 Easy

Topic: Sampling Methods

Learning Objective: 08-01 Explain why populations are sampled and describe four methods to sample a population.

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

4) When doing research, knowing the population mean and other population parameters is essential.

Answer: FALSE

Explanation: Often, the population mean and other population parameters are not known or available. Consequently, sample information that represents the population is used in research projects.

Difficulty: 2 Medium

Topic: Sampling Methods

Bloom’s: Understand

AACSB: Reflective Thinking

Accessibility: Keyboard Navigation

5) In stratified random sampling, a population is divided into strata using naturally occurring geographic or other boundaries. Then, strata are randomly selected and a random sample is collected from each strata.

Answer: FALSE

Explanation: In stratified sampling, the population is first divided into strata, and random samples are taken from each stratum. These strata represent portions of the population with identifiable characteristics, such as size or particular demographics. Dividing a population into naturally occurring geographic or other boundaries is known as cluster sampling.

Difficulty: 1 Easy

Topic: Sampling Methods

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

6) In cluster sampling, a population is divided into subgroups called clusters, and a sample is randomly selected from each cluster.

Answer: FALSE

Explanation: In cluster sampling, a random sample of the clusters is first taken, and then random samples are selected from each of the clusters. In this type of sampling, not all the clusters are used.

Difficulty: 1 Easy

Topic: Sampling Methods

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

7) The items or individuals of the population are arranged in a file drawer alphabetically by date received. A random starting point is selected and then every *k*th member of the population is selected for the sample. This sampling method is called systematic random sampling.

Answer: TRUE

Explanation: In systematic random sampling, the starting point is random and the selection of samples is done systematically (by every *k*th member) after that.

Difficulty: 1 Easy

Topic: Sampling Methods

Bloom’s: Remember

AACSB: Communication

Accessibility: Keyboard Navigation

8) When using stratified random sampling, the sampling error will be zero.

Answer: FALSE

Explanation: In all types of sampling, sampling error, the difference between the population value and the sample statistic, will occur.

Difficulty: 1 Easy

Topic: Sampling Error

Learning Objective: 08-02 Define sampling error.

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

9) If the size of a sample equals the size of the population, we would not expect any error in estimating the population parameter.

Answer: TRUE

Explanation: Sampling error occurs when one takes a sample from the population. If you do a census where all members of the population are included, there can be no sampling error.

Difficulty: 2 Medium

Topic: Sampling Error

Learning Objective: 08-02 Define sampling error.

Bloom’s: Apply

AACSB: Reflective Thinking

Accessibility: Keyboard Navigation

10) We can expect some difference between sample statistics and the corresponding population parameters. This difference is called the sampling error.

Answer: TRUE

Explanation: Sampling error is the difference between the sample statistic and the population parameter.

Difficulty: 1 Easy

Topic: Sampling Error

Learning Objective: 08-02 Define sampling error.

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

11) If 40 samples of size 21 were selected from a population of 22,493, we would expect the mean of the sample means and the population mean to be close but not exactly equal.

Answer: TRUE

Explanation: The mean of the sample means is exactly equal to the population mean when all possible samples of a given size are taken from that population. For a population of 22,493, far more than 40 possible samples of size 21 could be taken from the population, so while the mean of the sample means would be close to the population mean, it would not be exactly equal to the population mean due to sampling error.

Difficulty: 1 Easy

Topic: Sampling Error

Learning Objective: 08-02 Define sampling error.

Bloom’s: Understand

AACSB: Reflective Thinking

Accessibility: Keyboard Navigation

12) A sampling distribution of the means is a probability distribution consisting of all possible sample means of a given sample size selected from a population.

Answer: TRUE

Explanation: Sampling distributions show all possible sample means of a given size and the number of times each sample mean occurs for all possible samples taken from a population. This information is used to determine the probability of each possible sample mean occurring.

Difficulty: 1 Easy

Topic: Sampling Distribution of the Sample Mean

Learning Objective: 08-03 Demonstrate the construction of a sampling distribution of the sample mean.

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

13) If the sampling distribution of the sample means approximates a normal distribution, then the population must be normally distributed.

Answer: FALSE

Explanation: We cannot make any statements about the population distribution based on the sampling distribution of the sample means. However, the reverse is true: If the population is normal, then the distribution of the sample means will also be normal. If the population does not follow the normal distribution, the distribution of sample means will approach normal as the number of samples is increased.

Difficulty: 2 Medium

Topic: Sampling Distribution of the Sample Mean

Learning Objective: 08-03 Demonstrate the construction of a sampling distribution of the sample mean.

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

14) The mean of all possible sample means is equal to the population mean.

Answer: TRUE

Explanation: If all possible samples of a given size are taken from a population, and the mean of those sample means is calculated, it will be equal to the population mean.

Difficulty: 2 Medium

Topic: Sampling Distribution of the Sample Mean

Learning Objective: 08-03 Demonstrate the construction of a sampling distribution of the sample mean.

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

15) In the sampling distribution of the sample means, the standard error of the mean will vary according to the size of the sample. As the sample size, *n*, gets larger, the variability of the sampling distribution of the means gets larger.

Answer: FALSE

Explanation: As the size of the standard error of the mean, *σ*/, increases, the variability of the sampling distribution will decrease.

Difficulty: 2 Medium

Topic: Sampling Distribution of the Sample Mean

Bloom’s: Understand

AACSB: Communication

16) The central limit theorem states that for a sufficiently large sample, the sampling distribution of the means of all possible samples of size n generated from the population will be approximately normally distributed with the mean of the sampling distribution equal to *σ*2 and the variance equal to *σ*2/*n*.

Answer: FALSE

Explanation: The mean of the sample means will closely approximate the population mean, μ. The variance of the sampling distribution is *σ*2/*n*.

Difficulty: 2 Medium

Topic: The Central Limit Theorem

Learning Objective: 08-04 Recite the central limit theorem and define the mean and standard error of the sampling distribution of the sample mean.

Bloom’s: Understand

AACSB: Communication

17) The central limit theorem states that if the sample size, *n*, is sufficiently large, the sampling distribution of the means will be approximately normal, even when the population is skewed or uniform.

Answer: TRUE

Explanation: The central limit theorem shows that as the sample size increases, the sampling distribution of the sample mean will converge toward a normal distribution, even if the population itself is not normal.

Difficulty: 2 Medium

Topic: The Central Limit Theorem

Learning Objective: 08-04 Recite the central limit theorem and define the mean and standard error of the sampling distribution of the sample mean.

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

18) Based on the central limit theorem, the sample mean can be used as a good estimator of the population mean, assuming that the size of the sample is sufficiently large.

Answer: FALSE

Explanation: The results of the central limit theorem refer to the distribution of the sample means. It states that if the population is normal, the distribution of the sample means is also normal. If the population is not normal, then the distribution of the sample means will approach normal as we increase the sample size, *n*.

Difficulty: 1 Easy

Topic: The Central Limit Theorem

Learning Objective: 08-04 Recite the central limit theorem and define the mean and standard error of the sampling distribution of the sample mean.

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

19) For the sampling distribution of sample means, standard error will decrease as sample size increases.

Answer: TRUE

Explanation: As you increase sample size, the standard error of the mean decreases.

Difficulty: 3 Hard

Topic: The Central Limit Theorem

Bloom’s: Analyze

AACSB: Communication

Accessibility: Keyboard Navigation

20) The standard error of the mean is also called the sampling error.

Answer: FALSE

Explanation: Sampling error is the difference between a sample statistic and a population parameter, such as – *μ*.

Difficulty: 1 Easy

Topic: Standard Error of the Mean

Bloom’s: Remember

AACSB: Communication

21) The standard error of the mean measures the dispersion of the sampling distribution of the sample mean.

Answer: TRUE

Explanation: The standard error of the sampling mean, or *σ*/, measures the variability or dispersion of the sampling distribution of the sample mean.

Difficulty: 1 Easy

Topic: Standard Error of the Mean

Bloom’s: Remember

AACSB: Communication

22) The standard error of the mean is directly related to the sample size.

Answer: FALSE

Explanation: The standard error of the mean, *σ*/ , is indirectly or inversely related to the sample size.

Difficulty: 2 Medium

Topic: Standard Error of the Mean

Bloom’s: Understand

AACSB: Reflective Thinking

23) When computing probabilities for the sampling distribution of the sample mean, the *z*-statistic is computed as *z* = .

Answer: FALSE

Explanation: The denominator is the standard error of the mean (*σ*/) not the standard deviation (*σ*), so *z* =

Difficulty: 2 Medium

Topic: Using the Sampling Distribution of the Sample Mean

Learning Objective: 08-05 Apply the central limit theorem to calculate probabilities.

Bloom’s: Understand

AACSB: Communication

24) The central limit theorem allows us to use a *z*-statistic to compute probabilities for the sampling distribution of the sample mean.

Answer: TRUE

Explanation: The sampling distribution of the sampling mean will be normally distributed if the samples are taken from a normally distributed population or if the samples are taken from a population whose shape is not known or not normal, but the sample size is sufficiently large. If these conditions exist, then we can use the standard normal distribution to calculate probabilities.

Difficulty: 2 Medium

Topic: Using the Sampling Distribution of the Sample Mean

Learning Objective: 08-05 Apply the central limit theorem to calculate probabilities.

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

25) To study the population of consumer perceptions of new technology, sampling of the population is preferred over surveying the population because ________.

- A) sampling is more accurate
- B) we can compute
*z*-scores - C) it is quicker
- D) sampling methods are simple

Answer: C

Explanation: It will be quicker to collect information with a sample instead of from the entire population.

Difficulty: 1 Easy

Topic: Sampling Methods

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

26) When testing the safety of cars using crash tests, a sample of one or two cars is used because ________.

- A) sampling is more accurate
- B) cars are destroyed
- C) it is quicker
- D) the population is very large

Answer: B

Explanation: A very small sample would be used because the crash test will destroy the car.

Difficulty: 1 Easy

Topic: Sampling Methods

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

27) A marketing firm is studying consumer preferences for winter fashions in four different months. From a population of women 18 to 21 years of age, a random sample of 100 women was selected in January. Another random sample of 100 women was selected in March. Another random sample of 100 women was selected in June. Another random sample of 100 women was selected in September. What is the sample size?

- A) 4
- B) 100
- C) 400
- D) 1

Answer: B

Explanation: There are four samples of 100 each, so the sample size is 100.

Difficulty: 2 Medium

Topic: Sampling Methods

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

28) A marketing firm is studying consumer preferences for winter fashions in four different months. From a population of women 18 to 21 years of age, a random sample of 100 women was selected in January. Another random sample of 100 women was selected in March. Another random sample of 100 women was selected in June. Another random sample of 100 women was selected in September. What is the number of samples?

- A) 4
- B) 100
- C) 400
- D) 1

Answer: A

Explanation: There are four samples of 100 each, so the number of samples is four.

Difficulty: 2 Medium

Topic: Sampling Methods

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

29) When all the items in a population have an equal chance of being selected for a sample, the process is called ________.

- A) simple random sampling
- B) z-score
- C) sampling error
- D) nonprobability sampling

Answer: A

Explanation: In a simple random sample, each member of the population has the same chance of being selected for the sample.

Difficulty: 1 Easy

Topic: Sampling Methods

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

30) What is the difference between a sample mean and the population mean called?

- A) Standard error of the mean
- B) Sampling error
- C) Interval estimate
- D) Point estimate

Answer: B

Explanation: The sampling error is the difference between the population mean and the sample mean, – *μ*.

Difficulty: 1 Easy

Topic: Sampling Error

Learning Objective: 08-02 Define sampling error.

Bloom’s: Understand

AACSB: Communication

31) Suppose we select every fifth invoice in a file. What type of sampling is this?

- A) Random
- B) Cluster
- C) Stratified
- D) Systematic

Answer: D

Explanation: In a systematic sample, we pick a random starting point and then select every *k*th observation. For example, in a file of invoices, we would select a random invoice, and then select every fifth invoice.

Difficulty: 1 Easy

Topic: Sampling Methods

Bloom’s: Remember

AACSB: Communication

Accessibility: Keyboard Navigation

32) All possible samples of size *n* are selected from a population, and the mean of each sample is determined. What is the mean of the sample means?

- A) It is the population mean.
- B) It is larger than the population mean.
- C) It is smaller than the population mean.
- D) It cannot be estimated in advance.

Answer: A

Explanation: The mean of all sample means is exactly equal to the population mean.

Difficulty: 2 Medium

Topic: Sampling Distribution of the Sample Mean

Bloom’s: Understand

AACSB: Reflective Thinking

Accessibility: Keyboard Navigation

33) When dividing a population into subgroups so that a random sample from each subgroup can be collected, what type of sampling is used?

- A) Simple random sampling
- B) Systematic sampling
- C) Stratified random sampling
- D) Cluster sampling

Answer: C

Explanation: In stratified sampling, we divide the population into strata and then randomly select samples from each stratum.

Difficulty: 1 Easy

Topic: Sampling Methods

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

34) The true sampling error is usually not known because ________.

- A)
*µ*is unknown - B)
*µ*is a random variable - C)
*σ*2is unknown - D) The sample mean cannot be computed.

Answer: A

Explanation: Sampling error, – *μ*, is the difference between a sample mean and the population mean. Usually we do not know the population mean, so we cannot determine the sampling error.

Difficulty: 1 Easy

Topic: Sampling Error

Learning Objective: 08-02 Define sampling error.

Bloom’s: Understand

AACSB: Communication

35) The size of the sampling error is ________.

- A) directly related to the sample size—in other words, the larger the sample size, the larger the sampling error
- B) directly related to the population mean—in other words, the larger the mean, the larger the sampling error
- C) inversely related to the sample size—in other words, the larger the sample size, the smaller the sampling error
- D) inversely related to the population standard deviation—in other words, the smaller the standard deviation, the larger the sampling error

Answer: C

Explanation: The sampling error will be smaller with a larger sample size.

Difficulty: 3 Hard

Topic: Sampling Error

Learning Objective: 08-02 Define sampling error.

Bloom’s: Analyze

AACSB: Communication

Accessibility: Keyboard Navigation

36) The mean of all the sample means is ________.

A)

- B)
*µ* - C)
*σ* - D)
*α*

Answer: B

Explanation: The mean of all sample means is equal to the population mean.

Difficulty: 2 Medium

Topic: Sampling Distribution of the Sample Mean

Bloom’s: Understand

AACSB: Communication

37) For a given population, the mean of all the sample means , of sample size *n*, and the mean of all (*N*) population observations (*X*) are ________.

- A) equal to
- B) not equal
- C) equal to –
*μ* - D) equal to the population mean
*µ*

Answer: D

Explanation: The mean of the sample means is equal to the population mean.

Difficulty: 2 Medium

Topic: Sampling Distribution of the Sample Mean

Bloom’s: Understand

AACSB: Communication

38) Sampling error is defined as ________.

- A) –
*μ* - B)
*σ* - C)
*σ/n* - D)
*N − n*

Answer: A

Explanation: The sampling error is the difference between the sample mean and the population mean.

Difficulty: 2 Medium

Topic: Sampling Error

Learning Objective: 08-02 Define sampling error.

Bloom’s: Understand

AACSB: Communication

39) For a distribution of sample means constructed by sampling 5 items from a population of 15, ________.

- A) the sample size is 15
- B) there will be 3,003 possible sample means
- C) the mean of the sample means will be 3
- D) the standard error will be 1

Answer: B

Explanation: To determine the number of samples, we use the combination formula, where *n* is the sample size and *N* is the population size.

*NC**n* = 15*C*5 = = 3003

Difficulty: 2 Medium

Topic: Sampling Distribution of the Sample Mean

Bloom’s: Understand

AACSB: Communication

40) As the size of the sample increases, what happens to the shape of the distribution of sample means?

- A) It cannot be predicted in advance.
- B) It approaches a normal distribution.
- C) It is positively skewed.
- D) It is negatively skewed.

Answer: B

Explanation: When we increase the size of the sample, the sampling distribution of sample means approaches a normal distribution.

Difficulty: 2 Medium

Topic: The Central Limit Theorem

Bloom’s: Understand

AACSB: Communication

Accessibility: Keyboard Navigation

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