Deck 6: Sampling Distributions

Question 1

Sample statistics are random variables, because different samples can lead to different values of the sample statistics.

Answer 1

True

Question 2

In most situations, the true mean and standard deviation are unknown quantities that have to be estimated.

Answer 2

True

Question 3

A point estimator of a population parameter is a rule or formula which tells us how to use sample data to calculate a single number that can be used as an estimate of the population parameter.

Answer 3

True

Question 4

When estimating the population mean, the sample mean is always a better estimate than the sample median.

Question 5

A statistic is biased if the mean of the sampling distribution is equal to the parameter it is intended to estimate.

Question 6

Consider the population described by the probability distribution below.

Question 7

Consider the population described by the probability distribution below.

Question 8

The probability distribution shown below describes a population of measurements. Suppose that we took repeated random samples of n = 2 observations from the population described above. Which of the following would represent the sampling distribution of the sample mean?

Question 9

The sampling distribution of a sample statistic calculated from a sample of n measurements is the probability distribution of the statistic.

Question 10

Consider the probability distribution shown here. 6.2 Properties of Sampling Distributions: Unbiasedness and Minimum Variance 1 Understand Unbiasedness

Question 11

The sampling distribution of the sample mean is shown below. $$\begin{array} { c | c c c c c } \overline { \mathrm { x } } & 4 & 5 & 6 & 7 & 8 \\\hline \mathrm { p } ( \overline { \mathrm { x } } ) & 1 / 9 & 2 / 9 & 3 / 9 & 2 / 9 & 1 / 9\end{array}$$ Find the expected value of the sampling distribution of the sample mean.

A) 4
B) 5
C) 6
D) 7

Question 12

The probability distribution shown below describes a population of measurements. $$\begin{array} { c | c c c } \mathrm { x } & 0 & 2 & 4 \\\hline \mathrm { p } ( \mathrm { x } ) & 1 / 3 & 1 / 3 & 1 / 3\end{array}$$ Suppose that we took repeated random samples of n = 2 observations from the population described above. Find the expected value of the sampling distribution of the sample mean.

A) 0
B) 1
C) 2
D) 3
E) 4

Question 13

The probability distribution shown below describes a population of measurements that can assume values of 2, 5, 8, and 11, each of which occurs with the same frequency:

Question 14

If $\bar { x }$ is a good estimator for ?, then we expect the values of $\bar { x }$ to cluster around ?.

Question 15

The sample mean, $\bar { x }$ , is a statistic.

Question 16

The probability distribution shown below describes a population of measurements histogram for the sampling distribution of x.

Question 17

The term statistic refers to a population quantity, and the term parameter refers to a sample quantity.

Question 18

The length of time a traffic signal stays green (nicknamed the "green time") at a particular intersection follows a normal probability distribution with a mean of 200 seconds and the standard deviation of 10 seconds. Use this information to answer the following questions. Which of the following describes the derivation of the sampling distribution of the sample mean?

A) A single sample of sufficiently large size is randomly selected from the population of "green times" and its probability is determined.
B) The means of a large number of samples of size n randomly selected from the population of "green times" are calculated and their probabilities are plotted.
C) The mean and median of a large randomly selected sample of "green times" are calculated. Depending on whether or not the population of "green times" is normally distributed, either the mean or the median is
Chosen as the best measurement of center.
D) The standard deviations of a large number of samples of size n randomly selected from the population of "green times" are calculated and their probabilities are plotted.

Question 19

Which of the following describes what the property of unbiasedness means?

A) The sampling distribution in question has the smallest variation of all possible sampling distributions.
B) The center of the sampling distribution is found at the population standard deviation.
C) The center of the sampling distribution is found at the population parameter that is being estimated.
D) The shape of the sampling distribution is approximately normally distributed.

Question 20

The probability of success, p, in a binomial experiment is a parameter, while the mean and standard deviation, μ and σ, are statistics.

Question 21

Consider the population described by the probability distribution below.

Question 22

The ideal estimator has the greatest variance among all unbiased estimators.

Question 23

Consider the population described by the probability distribution below. a. Find μ. b. Find the sampling distribution of the sample mean for a random sample of n = 3 measurements from this distribution. c. Find the sampling distribution of the sample median for a random sample of n = 3 observations from this population. d. Show that both the mean and the median are unbiased estimators of μ for this population. e. Find the variances of the sampling distributions of the sample mean and the sample median. f. Which estimator would you use to estimate μ? Why? 6.3 The Sampling Distribution of x-bar and the Central Limit Theorem 1 Understand Central Limit Theorem

Question 24

The amount of time it takes a student to walk from her home to class has a skewed right distribution with a mean of 14 minutes and a standard deviation of 1.1 minutes. If times were collected from 40 randomly selected walks, describe the sampling distribution of , the sample mean time.

Question 25

The number of cars running a red light in a day, at a given intersection, possesses a distribution with a mean of 4.2 cars and a standard deviation of 6. The number of cars running the red light was observed on 100 randomly chosen days and the mean number of cars calculated. Describe the sampling distribution of the sample mean.

A) approximately normal with mean = 4.2 and standard deviation = 0.6
B) approximately normal with mean = 4.2 and standard deviation = 6
C) shape unknown with mean = 4.2 and standard deviation = 6
D) shape unknown with mean = 4.2 and standard deviation = 0.6

Question 26

The minimum-variance unbiased estimator (MVUE) has the least variance among all unbiased estimators.

Question 27

The Central Limit Theorem guarantees that the population is normal whenever n is sufficiently large.

Question 28

The daily revenue at a university snack bar has been recorded for the past five years. Records indicate that the mean daily revenue is $2700 and the standard deviation is $400. The distribution is skewed to the right due to several high volume days (football game days). Suppose that 100 days are randomly selected and the average daily revenue computed. Which of the following describes the sampling distribution of the sample mean?

A) normally distributed with a mean of $2700 and a standard deviation of $40
B) normally distributed with a mean of $2700 and a standard deviation of $400
C) normally distributed with a mean of $270 and a standard deviation of $40
D) skewed to the right with a mean of $2700 and a standard deviation of $400

Question 29

The Central Limit Theorem states that the sampling distribution of the sample mean is approximately normal under certain conditions. Which of the following is a necessary condition for the Central Limit Theorem to be used?

A) The sample size must be large .
B) The population from which we are sampling must be normally distributed.
C) The population size must be large .
D) The population from which we are sampling must not be normally distributed.

Question 30

A random sample of size n is to be drawn from a population with μ = 1500 and σ = 200. What size sample would be necessary in order to reduce the standard error to 20? 2 Find Mean, Standard Deviation

Question 31

As the sample size gets larger, the standard error of the sampling distribution of the sample mean gets larger as well.

Question 32

Suppose students' ages follow a skewed right distribution with a mean of 21 years old and a standard deviation of 2 years. If we randomly sample 450 students, which of the following statements about the sampling distribution of the sample mean age is incorrect?

A) The standard deviation of the sampling distribution is equal to 2 years.
B) The shape of the sampling distribution is approximately normal.
C) The mean of the sampling distribution is approximately 21 years old.
D) All of the above statements are correct.

Question 33

Which of the following does the Central Limit Theorem allow us to disregard when working with the sampling distribution of the sample mean?

A) The shape of the population distribution.
B) The mean of the population distribution.
C) The standard deviation of the population distribution.
D) All of the above can be disregarded when the Central Limit Theorem is used.

Question 34

The Central Limit Theorem is considered powerful in statistics because __________.

A) it works for any population distribution provided the sample size is sufficiently large
B) it works for any sample size provided the population is normal
C) it works for any population distribution provided the population mean is known
D) it works for any sample provided the population distribution is known

Question 35

The Central Limit Theorem is important in statistics because _____.

A) for a large n, it says the sampling distribution of the sample mean is approximately normal, regardless of the population
B) for a large n, it says the population is approximately normal
C) for any population, it says the sampling distribution of the sample mean is approximately normal, regardless of the sample size
D) for any size sample, it says the sampling distribution of the sample mean is approximately normal

Question 36

Suppose a random sample of n = 36 measurements is selected from a population with mean μ = 256 and variance

Question 37

Which of the following describes what the property of minimum variance means?

A) The sampling distribution in question has the smallest variation of all possible unbiased sampling distributions.
B) The center of the sampling distribution is found at the population standard deviation.
C) The center of the sampling distribution is found at the population parameter that is being estimated.
D) The shape of the sampling distribution is approximately normally distributed.

Question 38

Consider the population described by the probability distribution below.

Question 39

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

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

Question 40

The standard error of the sampling distribution of the sample mean is equal to σ, the standard deviation of the population.

Question 41

Suppose a random sample of n = 64 measurements is selected from a population with mean μ = 65 and and σ. standard deviation

Question 42

The weight of corn chips dispensed into a 15-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 15.5 ounces and a standard deviation of 0.1 ounce. Suppose 400 bags of chips are randomly selected. Find the probability that the mean weight of these 400 bags exceeds 15.6 ounces.

A) approximately 0
B) .1915
C) .3085
D) .6915

Question 43

The average score of all golfers for a particular course has a mean of 73 and a standard deviation of 3.5. Suppose 49 golfers played the course today. Find the probability that the average score of the 49 golfers exceeded 74.

A) .0228
B) .1293
C) .4772
D) .3707

Question 44

One year, the distribution of salaries for professional sports players had mean $1.8 million and standard deviation $0.6 million. Suppose a sample of 400 major league players was taken. Find the approximate probability that the average salary of the 400 players that year exceeded $1.1 million.

A) approximately 1
B) approximately 0
C) .2357
D) .7357

Question 45

Suppose a random sample of n = 64 measurements is selected from a population with mean μ = 65 and standard deviation 3 Find Probability

Question 46

Suppose a random sample of n = 64 measurements is selected from a population with mean μ = 65 and standard deviation

Question 47

The weight of corn chips dispensed into a 10-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 10.5 ounces and a standard deviation of .2 ounce. Suppose 100 bags of chips are randomly selected. Find the probability that the mean weight of these 100 bags exceeds 10.45 ounces.