Deck 10: Sampling and Sampling Distributions

Question 1

If the population is divided into relatively small subdivisions and then some of these subdivisions are selected for inclusion in the overall sample, the method is called _______ sampling.

A) stratified
B) cluster
C) quota
D) systematic

Answer 1

cluster

Question 2

The possible presence of hidden periodicities is a danger of _______ sampling.

A) stratified
B) cluster
C) simple random
D) systematic

Answer 2

systematic

Question 3

The standard error of the mean of a sample of 100 items from a population will be equal to _______ times the standard error of the mean of 25 items.

A) $\frac{1}{4}$
B) $\frac{1}{2}$
C) 4
D) 2

Answer 3

$\frac{1}{2}$

Question 4

The value we use for the standard error of the mean when $n$ is at least $5 \%$ of the population size is

A) $\frac{\sigma}{\sqrt{n}} \sqrt{\frac{N-n}{N-1}}$ .
B) $\sigma$ .
C) $\frac{\sigma}{\sqrt{n}}$ .
D) $s$ .

Question 5

If the digits $0,2,4,6$ , and 8 represent heads and the digits 1, 3, 5, 7, and 9 represent tails, then the random numbers 527, 648, 910, and 485 represent what sequence of numbers of heads in four flips of three coins?

A) $2,1,1,3$
B) $2,2,1,3$
C) $2,0,2,1$
D) $1,3,1,2$

Question 6

Suppose we obtain a sample mean of 80 from a sample of size 100 from an infinite population with $\sigma=50$ . Then the probability is 0.75 that the error in estimating the population mean using Chebyshev's theorem is:

A) between 30 and 130 .
B) less than 6.
C) between 74 and 86 .
D) less than 10 .

Question 7

Our sample size criterion for applying the central limit theorem if the population may not be normal is

A) $n \geq 30$ .
B) $n \geq 0.05 \mathrm{~N}$ .
C) $n<0.05 \mathrm{~N}$ .
D) $n \geq 5$ .

Question 8

The central limit theorem can be used to estimate the

A) population mean from the sample mean.
B) probability of error in using $\bar{x}$ to estimate $\mu$ .
C) mean of the sampling distribution by using the population mean.
D) standard error of the mean by using $\sigma$ .

Question 9

An important goal of _______ sampling is that there is less variability within the resulting subgroups than in the entire population.

A) stratified
B) cluster
C) quota
D) systematic

Question 10

A stratified sample is taken from a population of size $N=2000$ which consists of two strata of $N_{1}=800$ , $N_{2}=1200, n_{1}=20, n_{2}=40, \sigma_{1}=3, \sigma_{2}=5$ . Then the method of allocation is

A) proportional.
B) optimum.
C) neither proportional nor optimum.
D) both proportional and optimum.

Question 11

If the mean of a random sample of size 400 is used to estimate the mean of an infinite population with a standard deviation of 60 , the probability is 0.95 that the error is less than

A) 1.96 .
B) 117.6 .
C) 3.92 .
D) 5.88 .

Question 12

If we want to change a standard error from 12 to 4 by changing the sample size, we need to multiply the original sample size by

A) 3 .
B) 9 .
C) $\frac{1}{3}$ .
D) $\frac{1}{9}$ .

Question 13

Write the word or phrase that best completes each statement or answers the question.

-How many different samples of size 2 can be selected from a population of size 15 ?

Question 14

Write the word or phrase that best completes each statement or answers the question.

-How many different samples of size 3 can be selected from a population of 12 ?

Question 15

An investor considers investing in the stocks of the following companies: IBM, Honeywell, Mobil, Eastman Kodak, and Homestake Mining.
-In how many ways can the investor select two stocks to buy?

Question 16

An investor considers investing in the stocks of the following companies: IBM, Honeywell, Mobil, Eastman Kodak, and Homestake Mining.
-In how many ways can the investor select three stocks to buy?

Question 17

An investor considers investing in the stocks of the following companies: IBM, Honeywell, Mobil, Eastman Kodak, and Homestake Mining.
-Find the probability of selecting any particular sample of two stocks.

Question 18

An investor considers investing in the stocks of the following companies: IBM, Honeywell, Mobil, Eastman Kodak, and Homestake Mining.
-Find the probability of any particular one of the companies being included in a sample of two stocks.

Question 19

Random samples of size 2 are selected from the finite population which consists of the numbers 4, 7, 10, 13, 16, and 19.
-Find the mean and standard deviation of this population.

Question 20

Random samples of size 2 are selected from the finite population which consists of the numbers 4, 7, 10, 13, 16, and 19.

-List the 15 possible random samples of size 2 that can be selected from the finite population, and calculate their means.

Question 21

Random samples of size 2 are selected from the finite population which consists of the numbers 4, 7, 10, 13, 16, and 19.

-Construct the sampling distribution of the mean for the 15 possible random samples of size 2 from the given finite population. Assign each possible sample a probability of $\frac{1}{15}$ .

Question 22

Random samples of size 2 are selected from the finite population which consists of the numbers 4, 7, 10, 13, 16, and 19.
-Calculate the mean and variance of the probability distribution of the mean for the 15 possible random samples of size 2 from the given finite population. Verify the results by comparing them with the results obtained by using the appropriate formulas.

Question 23

Suppose that in a group of six athletes there are three jockeys whose heights are 60,65 , and 55 inches, and three basketball players whose heights are 80,75 , and 85 inches.

-List all possible random samples of size 2 which may be taken from this population. Calculate the means of these samples, and calculate $\sigma_{\bar{x}}$ .

Question 24

Suppose that in a group of six athletes there are three jockeys whose heights are 60,65 , and 55 inches, and three basketball players whose heights are 80,75 , and 85 inches.

-List all possible stratified random samples of size 2 which may be taken by selecting one jockey and one basketball player. Calculate the means of these samples, and calculate $\sigma_{\bar{x}}$ .

Question 25

Suppose that in a group of six athletes there are three jockeys whose heights are 60,65 , and 55 inches, and three basketball players whose heights are 80,75 , and 85 inches.

-Suppose that the six athletes are divided into clusters according to their sports, each cluster is assigned a probability of $\frac{1}{2}$ , and a random sample of size 2 is taken from one of the randomly chosen clusters. List all possible samples, calculate their means, and calculate $\sigma_{\bar{x}}$ .

Question 26

Suppose we want to estimate the mean salary of seven people based on a sample of size 3 . The salaries of the seven people in thousands of dollars are $25,10,19,57,53,55$ , and 63 , so that the population means we want to estimate is $\mu=36.5$ . If the first three of these salaries are those of bank trainees and the other four are salaries of officers in the bank, find $n_{1}$ and $n_{2}$ if the allocation is
-proportional.

Question 27

Suppose we want to estimate the mean salary of seven people based on a sample of size 3 . The salaries of the seven people in thousands of dollars are $25,10,19,57,53,55$ , and 63 , so that the population means we want to estimate is $\mu=36.5$ . If the first three of these salaries are those of bank trainees and the other four are salaries of officers in the bank, find $n_{1}$ and $n_{2}$ if the allocation is
-optimum.

Question 28

A stratified sample of size $n=300$ is to be taken from a population of size $N=30,000$ which consists of four strata for which $N_{1}=10,000, N_{2}=5,000, N_{3}=8,000, N_{4}=7,000, \sigma_{1}=15, \sigma_{2}=25, \sigma_{3}=20$ and $\sigma_{4}=30$ . How large a sample must be taken from each stratum if the allocation is to be
-proportional?

Question 29

A stratified sample of size $n=300$ is to be taken from a population of size $N=30,000$ which consists of four strata for which $N_{1}=10,000, N_{2}=5,000, N_{3}=8,000, N_{4}=7,000, \sigma_{1}=15, \sigma_{2}=25, \sigma_{3}=20$ and $\sigma_{4}=30$ . How large a sample must be taken from each stratum if the allocation is to be
-optimum?

Question 30

When we sample from an infinite population, what happens to the standard error of the mean if the sample size is increased from 64 to 400 ?

Question 31

When we sample from an infinite population, what happens to the standard error of the mean if the sample size is decreased from 225 to 100 ?

Question 32

The mean number of errors in a random sample of 100 accounts to be audited is used to estimate the mean of the population of accounts having a standard deviation of $\sigma=4$ . What is the probability that the error will be less than 0.8
-using Chebyshev's theorem?

Question 33

The mean number of errors in a random sample of 100 accounts to be audited is used to estimate the mean of the population of accounts having a standard deviation of $\sigma=4$ . What is the probability that the error will be less than 0.8
-using the central limit theorem?

Question 34

The mean of a random sample of size 49 is used to estimate the mean of a very large population, consisting of the lifetimes of certain stereo components which have a standard deviation of $\sigma=35$ hours. What is the probability that our estimate will be in error by
-less than 8 hours?

Question 35

The mean of a random sample of size 49 is used to estimate the mean of a very large population, consisting of the lifetimes of certain stereo components which have a standard deviation of $\sigma=35$ hours. What is the probability that our estimate will be in error by
-less than 6 hours?

Question 36

A new cure for the common cold has the side effect of producing dizziness in the user for an average of 40 minutes with a standard deviation of 12 minutes.
-In a random sample of 64 people who take the drug, find the probability that the average time that a user remains dizzy is at least 38 minutes.

Question 37

A new cure for the common cold has the side effect of producing dizziness in the user for an average of 40 minutes with a standard deviation of 12 minutes.
-In a random sample of 64 people who take the drug, find the probability that the average time that a user remains dizzy is between 41 and 45 minutes.

Question 38

A new cure for the common cold has the side effect of producing dizziness in the user for an average of 40 minutes with a standard deviation of 12 minutes.
-In a random sample of 64 people who take the drug, give the standard error of the mean for the random variable.

Question 39

Using the population of the numbers $10,38,25,18,42,15$ , construct the sampling distribution of the median for random samples of size $n=3$ drawn without replacement from the given population.

Question 40

The weights of ice cream in "one-scoop" cones by an employee of a certain ice cream store is known to have a mean of 4 ounces with a standard deviation of . 6 ounces. What is the probability that in a random sample of 36 "one -scoop" cones the average weight is
-between 3.9 and 4.2 ounces?

Question 41

The weights of ice cream in "one-scoop" cones by an employee of a certain ice cream store is known to have a mean of 4 ounces with a standard deviation of . 6 ounces. What is the probability that in a random sample of 36 "one -scoop" cones the average weight is
-at least 3.7 ounces?

Question 42

A simple random sample from a finite population is a sample which is chosen in such a way that each possible sample has the same probability of being selected.

Question 43

Sampling with replacement from a finite population is, in effect, sampling from an infinite population.

Question 44

When sampling with replacement, the finite population correction factor can be omitted.

Question 45

In order for $\sigma_{\bar{\chi}}^{\bar{x}}$ to be as low as possible in stratified sampling, it is preferable that each of the strata have as little homogeneity as possible.

Question 46

The method of allocation such that for a fixed size, the sample chosen will have the smallest possible standard error for the estimate of the population mean is called optimum allocation.

Question 47

A judgment sample is usually not a random sample.

Question 48

A sampling distribution of the mean is always a probability distribution whose values are sample means.

Question 49

It is impossible to apply the central limit theorem if the population does not follow a normal distribution.

Question 50

The variance of the sampling distribution can be equal to the variance of the population.

Question 51

The central limit theorem applies in situations when $n$ constitutes a large proportion of the population.

Question 52

We use the $\frac{\sigma}{\sqrt{n}}$ for the standard error of the mean if the sample size is _______.

Question 53

A random sample of size 36 is selected from a population of size 101 whose standard deviation is 5 . The standard error of the mean is _______.

Question 54

The allocation in stratified sampling will be proportional if $n_{i}=$ _______ for $i=1,2, \ldots, k$ .

Question 55

If the population is divided into three strata and $\frac{n_{1}}{N_{1} \sigma_{1}}=\frac{n_{2}}{N_{2} \sigma_{2}}=\frac{n_{3}}{N_{3} \sigma_{3}}$ , then the allocation is called _______

Question 56

Using simple random sampling, the probability of selecting any particular sample of size 3 from a population of size 30 is _______.

Question 57

The central limit theorem is important since it justifies the use of _______ in solving problems.

Question 58

The distribution of the totality of all sample means is called _______.

Question 59

The error in estimating the mean of a population by using a sample mean is expressed in symbols by _______.

Question 60

If we sample with replacement from a finite population, our sample would be random if in each draw all elements of the population _______ and successive draws are _______.

Question 61

The central limit theorem gives the probability that the will be less than a given number when we use the mean of a sample to estimate the _______.