Question 1

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, not a cluster sample. In a cluster sample, the population is divided into groups, or clusters, and then a random sample of clusters is selected. Each individual in the selected clusters would be included in the sample.

Question 2

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 a sample of any size, it says the sampling distribution of the sample mean is approximately normal.

Accepted Answer

The Central Limit Theorem says that for a large n, the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population. This is important because it allows us to use techniques that rely on the assumption of normality, even when we don't know the shape of the underlying population.

Question 3

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 4

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 three 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

This is an example of cluster sampling because the population (students in the class) was divided into clusters (left, right, center) and one cluster was randomly selected for the sample.

Question 5

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 6

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 of the sample mean is always equal to the population mean μ, regardless of the sample size or the population distribution. However, the standard deviation of the sampling distribution is not always equal to the population standard deviation σ. It depends on the sample size and the population distribution. Finally, the shape of the sampling distribution is not always approximately normal, but it becomes more normal as the sample size increases, due to the Central Limit Theorem.

Question 7

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 states that as the sample size increases, the sampling distribution of the mean approaches a normal distribution regardless of the shape of the population. This is particularly useful when dealing with large sample sizes because it allows us to assume normality even if the population is not normally distributed.

Question 8

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, which are calculated from sample data. They do not describe the distribution of parameters, which are numerical characteristics of the population that the sample is drawn from.

Question 9

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 answer of Suppose the ages of students in Statistics...

Question 10

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 instead equal to σ/√n where σ is the population standard deviation and n is the sample size.

Question 11

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 answer of Suppose a sample of n = 50...

Question 12

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 for which the expected value is equal to the population parameter it is estimating. This means that on average, the sample statistic will provide an accurate estimate of the population parameter. A biased sample statistic, on the other hand, would be systematically overestimating or underestimating the population parameter. Minimum variance and random are not directly related to the concept of bias in statistics.

Question 13

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 14

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

Accepted Answer

Asking for a rounded figure can lead to measurement error as the actual earning may not precisely align with the rounded figure provided by the respondent. Additionally, the respondent may also provide an estimate instead of the precise figure, causing further measurement error.

Question 15

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 16

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

Accepted Answer

The question is asking about illicit drug use over the past two months, which is a sensitive and potentially embarrassing topic. It is possible that someone may be hesitant or afraid to answer honestly, leading to measurement error. Additionally, the question does not specify which types of drugs are considered illicit, which could lead to confusion or differing interpretations of the question.

Question 17

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 important to report when presenting research findings, but it becomes an ethical issue when it is purposely hidden or ignored to promote a certain viewpoint or agenda. Failing to report sample size and margin of error can mislead the public and decision-makers into believing a finding is more significant or generalizable than it actually is.

Question 18

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

Accepted Answer

Selection bias is more likely in systematic sampling because the method involves selecting elements from a list at regular intervals, which can inadvertently favor certain patterns within the population. In contrast, simple random sampling, where every member of the population has an equal chance of being selected, minimizes the risk of such bias.

Question 19

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 standard error of the mean is always less than or equal to the population standard deviation, decreases with larger sample sizes, and measures the variability of sample means.

Question 20

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...

Quiz 7: Sampling and Sampling Distributions

The Central Limit Theorem is important in statistics because

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 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.

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

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

Sampling distributions describe the distribution of

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?

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