Question 1

Sampling error becomes an ethical issue if the findings are purposely presented without reference to sample size and margin of error so that the sponsor can promote a viewpoint that might otherwise be truly insignificant.

Accepted Answer

Sampling error is always a concern in research, but it becomes an ethical issue if the findings are presented without reference to sample size and margin of error in order to promote a particular viewpoint. This can be seen as deliberately misleading or manipulative, and can undermine the integrity of the research and the trustworthiness of the sponsor. It is important for researchers and sponsors to be transparent about the limitations of their findings and to provide context and explanation for their results.

Question 2

The Central Limit Theorem is considered powerful in statistics because it works for any population distribution provided the sample size is sufficiently large and the population mean and standard deviation are known.

Accepted Answer

The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean will approach a normal distribution as the sample size becomes large, regardless of the shape of the population distribution, provided the population has a finite mean and standard deviation. This makes it a powerful tool in statistics for making inferences about population parameters.

Question 3

Why is the Central Limit Theorem so important to the study of sampling distributions?
A) It allows us to disregard the size of the sample selected when the population is not normal.
B) It allows us to disregard the shape of the sampling distribution when the size of the population is large.
C) It allows us to disregard the size of the population we are sampling from.
D) It allows us to disregard the shape of the population when n is large.

Accepted Answer

The Central Limit Theorem is important to the study of sampling distributions because it allows us to disregard the shape of the population when the sample size is large (usually n > 30). This means that even if the population is not normally distributed, the sampling distribution of the mean will still have an approximation to the normal distribution. This is a critical assumption in many statistical methods and models.

Question 4

Which of the following is true about the sampling distribution of the sample mean?
A) The mean of the sampling distribution is always μ.
B) The standard deviation of the sampling distribution is always σ.
C) The shape of the sampling distribution is always approximately normal.
D) All of the above are true.

Accepted Answer

The mean of the sampling distribution is always equal to the population mean, μ.

Answer: B
 The standard deviation of the sampling distribution is equal to the population standard deviation divided by the square root of the sample size, σ/√n.

Answer: C
 The shape of the sampling distribution is approximately normal if the sample size is large enough (typically n ≥ 30) or if the population distribution is normal. Therefore, this statement is not always true, but is often true in practice.

Therefore, the answer is: A, B.

Question 5

The professor of a business statistics class wanted to find out the mean amount of time per week her students spent studying for the class. She divided the fifty students on her roster into ten groups starting from the first student on the roster. The first student was randomly selected from the first group. Then every tenth student was selected from the remaining students. This is an example of a cluster sample.

Accepted Answer

This is an example of a systematic sample because every tenth student was selected after the first student was chosen randomly, not a cluster sample where entire groups or clusters are chosen at random.

Question 6

Which of the following statements about the sampling distribution of the sample mean is incorrect?
A) The sampling distribution of the sample mean is approximately normal whenever the sample size is sufficiently large (n ≥ 30).
B) The sampling distribution of the sample mean is generated by repeatedly taking samples of size n and computing the sample means.
C) The mean of the sampling distribution of the sample mean is equal to μ.
D) The standard deviation of the sampling distribution of the sample mean is equal to σ.

Accepted Answer

The standard deviation of the sampling distribution of the sample mean is not equal to σ, but rather to σ/√n.

Question 7

If the expected value of a sample statistic is equal to the parameter it is estimating, then we call that sample statistic
A) unbiased.
B) minimum variance.
C) biased.
D) random.

Accepted Answer

An unbiased sample statistic is one whose expected value is equal to the parameter it is estimating. This means that, on average, the statistic will provide an accurate estimate of the true value of the parameter. Both options B and D are not accurate definitions of an unbiased statistic. Option C, biased, is the opposite of unbiased, so it is not the correct choice.

Question 8

A sample that does not provide a good representation of the population from which it was collected is referred to as a(n) ________ sample.

Accepted Answer

The answer of A sample that does not provide a...

Question 9

For air travelers, one of the biggest complaints is of the waiting time between when the airplane taxis away from the terminal until the flight takes off. This waiting time is known to have a right skewed distribution with a mean of 10 minutes and a standard deviation of 8 minutes. Suppose 100 flights have been randomly sampled. Describe the sampling distribution of the mean waiting time between when the airplane taxis away from the terminal until the flight takes off for these 100 flights.
A) Distribution is right skewed with mean = 10 minutes and standard error = 0.8 minutes.
B) Distribution is right skewed with mean = 10 minutes and standard error = 8 minutes.
C) Distribution is approximately normal with mean = 10 minutes and standard error = 0.8 minutes.
D) Distribution is approximately normal with mean = 10 minutes and standard error = 8 minutes.

Accepted Answer

The Central Limit Theorem states that the sampling distribution of the sample mean will be approximately normal if the sample size is large enough, regardless of the shape of the population distribution. Here, with a sample size of 100, the distribution of the sample mean is approximately normal. The mean of the sampling distribution is the same as the population mean (10 minutes), and the standard error is calculated as the population standard deviation divided by the square root of the sample size, which is $8 / \sqrt{100} = 0.8$ minutes.

Question 10

The professor of a business statistics class wanted to find out the mean amount of time per week her students spent studying for the class. She divided the students into the left, right and center groups according to the location they sat in the class that day. One of these 3 groups was randomly selected and everyone in the group was asked the mean amount of time per week he/she spent studying for the class. This is an example of a cluster sample.

Accepted Answer

In a cluster sample, the population is divided into clusters and a random sample of clusters is selected. In this scenario, the professor divided the students into clusters according to their seating arrangement and randomly selected one cluster to sample from. This fits the definition of a cluster sample.

Question 11

The question: "How much did you earn last year rounded to the nearest hundreds of dollars?" will most likely result in measurement error.

Accepted Answer

Since the question asks the respondent to round their earnings to the nearest hundreds of dollars, it introduces the possibility of measurement error. The true earnings may not fall exactly on a multiple of 100, which could lead to rounding up or down and result in inaccuracies.

Question 12

The amount of time it takes to complete an examination has a left skewed distribution with a mean of 65 minutes and a standard deviation of 8 minutes. If 64 students were randomly sampled, the probability that the sample mean of the sampled students exceeds 71 minutes is approximately 0.

Accepted Answer

The answer of The amount of time it takes to...

Question 13

The average score of all pro golfers for a particular course has a mean of 70 and a standard deviation of 3.0. Suppose 36 pro golfers played the course today. Find the probability that the average score of the 36 pro golfers exceeded 71.

Accepted Answer

The answer of The average score of all pro golfers...

Question 14

The Central Limit Theorem is important in statistics because
A) for a large n, it says the population is approximately normal.
B) for any population, it says the sampling distribution of the sample mean is approximately normal, regardless of the sample size.
C) for a large n, it says the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population.
D) for any sized sample, it says the sampling distribution of the sample mean is approximately normal.

Accepted Answer

The Central Limit Theorem states that for a large enough sample size (n), the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population. This makes it a valuable tool in statistical analysis, as it allows for the use of normal distribution-based methods when analyzing sample means.

Question 15

Sampling distributions describe the distribution of
A) parameters.
B) statistics.
C) both parameters and statistics.
D) neither parameters nor statistics.

Accepted Answer

Sampling distributions describe the distribution of statistics, not parameters. Parameters refer to population characteristics that are usually unknown and fixed, while statistics refer to samples and are subject to variation.

Question 16

Selection bias occurs more frequently in systematic samples than in simple random samples.

Accepted Answer

Systematic sampling involves selecting every kth element from a list or population, which can introduce bias if the list does not have a random order or if there is a pattern in the list. In simple random sampling, each element has an equal chance of being selected, which reduces the likelihood of selection bias.

Question 17

Suppose the ages of students in Statistics 101 follow a right skewed distribution with a mean of 23 years and a standard deviation of 3 years. If we randomly sampled 100 students, which of the following statements about the sampling distribution of the sample mean age is incorrect?
A) The mean of the sampling distribution is equal to 23 years.
B) The standard deviation of the sampling distribution is equal to 3 years.
C) The shape of the sampling distribution is approximately normal.
D) The standard error of the sampling distribution is equal to 0.3 years.

Accepted Answer

The standard deviation of the sampling distribution (standard error) is the population standard deviation divided by the square root of the sample size, which is 0.3 years, not 3 years.

Question 18

The standard error of the mean
A) is never larger than the standard deviation of the population.
B) decreases as the sample size increases.
C) measures the variability of the mean from sample to sample.
D) All of the above.

Accepted Answer

The answer of The standard error of the mean
A) is...

Question 19

The question: "Have you used any form of illicit drugs over the past 2 months?" will most likely result in measurement error if the question is answered.

Accepted Answer

The question is asking about past behavior, which can be affected by recall bias. Additionally, the term "illicit drugs" may be interpreted differently by different people, leading to response bias. Overall, there is a high likelihood of measurement error with this question.

Question 20

Suppose a sample of n = 50 items is selected from a population of manufactured products and the weight, X, of each item is recorded. Prior experience has shown that the weight has a probability distribution with μ = 6 ounces and σ = 2.5 ounces. Which of the following is true about the sampling distribution of the sample mean if a sample of size 15 is selected?
A) The mean of the sampling distribution is 6 ounces.
B) The standard deviation of the sampling distribution is 2.5 ounces.
C) The shape of the sampling distribution is approximately normal.
D) All of the above are correct.

Accepted Answer

The mean of the sampling distribution of the sample mean is equal to the population mean, which is 6 ounces. The standard deviation of the sampling distribution is not 2.5 ounces; it is the population standard deviation divided by the square root of the sample size (σ/√n). The shape of the sampling distribution is approximately normal only if the sample size is large enough (typically n ≥ 30), so with n = 15, we cannot assume normality unless the population distribution is normal.

Quiz 7: Sampling and Sampling Distributions

Sampling error becomes an ethical issue if the findings are purposely presented without reference to sample size and margin of error so that the sponsor can promote a viewpoint that might otherwise be truly insignificant.

The Central Limit Theorem is considered powerful in statistics because it works for any population distribution provided the sample size is sufficiently large and the population mean and standard deviation are known.

Why is the Central Limit Theorem so important to the study of sampling distributions?

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

Which of the following statements about the sampling distribution of the sample mean is incorrect?

If the expected value of a sample statistic is equal to the parameter it is estimating, then we call that sample statistic

A sample that does not provide a good representation of the population from which it was collected is referred to as a(n) ________ sample.

The question: "How much did you earn last year rounded to the nearest hundreds of dollars?" will most likely result in measurement error.

The amount of time it takes to complete an examination has a left skewed distribution with a mean of 65 minutes and a standard deviation of 8 minutes. If 64 students were randomly sampled, the probability that the sample mean of the sampled students exceeds 71 minutes is approximately 0.

The average score of all pro golfers for a particular course has a mean of 70 and a standard deviation of 3.0. Suppose 36 pro golfers played the course today. Find the probability that the average score of the 36 pro golfers exceeded 71.

The Central Limit Theorem is important in statistics because

Sampling distributions describe the distribution of

Selection bias occurs more frequently in systematic samples than in simple random samples.

Suppose the ages of students in Statistics 101 follow a right skewed distribution with a mean of 23 years and a standard deviation of 3 years. If we randomly sampled 100 students, which of the following statements about the sampling distribution of the sample mean age is incorrect?

The standard error of the mean

The question: "Have you used any form of illicit drugs over the past 2 months?" will most likely result in measurement error if the question is answered.