# Sampling, central limit theorem, normal distribution

Question 1

All of the following are reasons to sample except for:

a) Reduce sampling error

b) The research process is sometimes destructive.

c) Save time

d) For given resources, broaden the scope of the study.

Question 2

Sampling in which every unit of the population has the same probability of being selected into the sample is sometimes referred to as:

a) representative sampling

b) fair sampling

c) random sampling

d) nonrandom sampling

Question 3

The most elementary random sampling technique is:

a) simple random sampling

b) stratified random sampling

c) systematic random sampling

d) area random sampling

Question 4

In which of the following sampling techniques does the researcher number every item of the population before taking the sample?

a) systematic sampling

b) quota sampling

c) cluster sampling

d) simple random sampling

Question 5

The main reason for using stratified random sampling is to:

a) reduce costs

b) make certain that every kth item is selected

c) reduce sampling error

d) reduce non sampling error

Question 6

Fifty percent of a population possesses attribute X, thirty percent possesses attribute Y, and twenty percent possesses attribute Z. A researcher decides to include some people with attribute X, some with attribute Y, and some with attribute Z in her sample. Her sample consists of 80 people with attribute X, 70 people with attribute Y, and 50 people with attribute Z. The researcher has most likely done what type of sampling?

a) systematic sampling

b) simple random sampling

c) proportionate stratified sampling

d) disproportionate stratified sampling

Question 7

Another name for cluster sampling is:

a) stratified sampling

b) quota sampling

c) snowball sampling

d) area sampling

Question 8

Test markets are probably closest to which type of sampling?

a) cluster sampling

b) quota sampling

c) stratified sampling

d) snowball sampling

Question 9

In using judgment sampling, the researcher attempts to sample elements from the population by using her judgment. However, the researcher tends to make errors of judgment in one direction. These systematic errors are called:

a) consistencies

b) directional errors

c) biases

d) tendencies

Question 10

Which of the following sampling techniques is based on referral?

a) stratified

b) quota

c) area

d) snowball

Question 11

The central limit theorem states that which of the following is true:

Question 12

The central limit theorem states that for a given large sample size, if the shape of the population is unknown, the distribution of sample means is:

a) unknown

b) normal

c) platykurtic

d) skewed

Question 13

Suppose a population has a mean of 75 and a standard deviation of 14. If a researcher randomly samples 35 values from this population, the probability that >= 72 is:

a) 0.102

b) 0.5832

c) 0.398

d) 0.898

Question 14

Suppose a population has a mean of 84 and a standard deviation of 18. If a researcher randomly samples 37 values from this population, the probability that 80 <= <= 89 is:

a) 0.866

b) 0.043

c) 0.1331

d) 0.3669

Question 15

Suppose a population has a mean of 152 and a standard deviation of 22. If a researcher is randomly taking samples of size 42 from the population, 63% of the sample means are greater than what value?

a) 153.12

b) 151.56

c) 150.88

d) 154.64

Question 16

The Central Limit Theorem applies to sample proportions if sample size is large enough relative to the population proportion. How large of a sample size is needed?

a) n? p = .25

b) n? p> 5 and n? q> 5

c) n >= 30

d) n? p> 7 and n< 20

Question 17

Fifty-seven percent of the population has heard of brand x batteries. If 340 people are randomly selected from the population, what is the probability that the sample proportion who have heard of brand x batteries is greater than sixty percent?

a) 0.1314

b) 0.2755

c) 0.3686

d) 0.8686

Question 18

Suppose that 78% of all prerecorded music shoppers are under age 30. If a random sample of 250 prerecorded music shoppers is randomly taken, what is the probability that the sample proportion that is under age 30 is more than 76%?

a) 0.2764

b) 0.7764

c) 0.2236

d) 0.8131

Question 19

Suppose that 42% of all consumers who purchase bottled water from a supermarket prefer brand x. If a random sample of 425 such consumers is taken, what is the probability that between 35% and 38% prefer brand x?

a) 0.0457

b) 0.1056

c) 0.3944

d) 0.9507

Question 20

Suppose .27 of all workers would switch jobs if they had an opportunity. If 292 workers are randomly selected, what is the probability that between 82 and 90 would switch jobs if they had an opportunity?

a) 0.1628

b) 0.2651

c) 0.4279

d) 0.5907

https://brainmass.com/statistics/normal-distribution/sampling-central-limit-theorem-normal-distribution-101701

#### Solution Preview

Please see the attached file.

Question 1 Multiple Choice

All of the following are reasons to sample except for:

Reduce sampling error

The research process is sometimes destructive.

Save time

For given resources, broaden the scope of the study.

Answer: Reduce sampling error

Question 2 Multiple Choice

Sampling in which every unit of the population has the same probability of being selected into the sample is sometimes referred to as:

representative sampling

fair sampling

random sampling

nonrandom sampling

Answer: random sampling

Question 3 Multiple Choice

The most elementary random sampling technique is:

simple random sampling

stratified random sampling

systematic random sampling

area random sampling

Answer: simple random sampling

Question 4 Multiple Choice

In which of the following sampling techniques does the researcher number every item of the population before taking the sample?

systematic sampling

quota sampling

cluster sampling

simple random sampling

Answer: systematic sampling

Question 5 Multiple Choice

The main reason for using stratified random sampling is to:

reduce costs

make certain that every kth item is selected

reduce sampling error

reduce non sampling error

Answer: reduce costs

Question 6 Multiple Choice

Fifty percent of a population possesses attribute X, thirty percent possesses attribute Y, and twenty percent possesses attribute Z. A researcher decides to include some people with attribute X, some with attribute Y, and some with attribute Z in her sample. Her sample consists of 80 people with attribute X, 70 people with attribute Y, and 50 people with attribute Z. The researcher has most likely done what type of sampling?

systematic sampling

simple random sampling

proportionate stratified sampling

disproportionate stratified sampling

Answer: disproportionate stratified sampling

X= 80 40% 50%

Y= 70 35% 30%

Z= 50 25% 20%

200

Therefore, disproportionate stratified sampling

Question 7 Multiple Choice

Another name for cluster sampling is:

stratified sampling

quota sampling

snowball sampling

area ...

#### Solution Summary

Answers Multiple choice questions on sampling, central limit theorem, normal distribution

Probability: Normal Distribution and the Central Limit Theorem

1. The weight of potato chips in a small size bag is stated to be 5 ounces. The amount that the packaging machine puts in these bags is believed to have a normal model with a mean of 5.1 ounces and a standard deviation of 0.08 ounces.

a) What fraction of all bags sold are underweight?

b) Some of the chips are sold in "bargain packs" of 3 bags. What's the probability that none of the 3 is underweight?

c) What's the probability that the mean weight of the 3 bags is below the stated amount?

d) What's the probability that the mean weight of a 24-bag case of potato chips is below 5oz?

2. Suppose that the IQ's of university A's students can be described by a normal model with mean 140 and standard deviation 7 points. Also suppose that IQ of students from university B can be described by a normal model with mean 110 and standard deviation 9.

a) The probability is __________.

b) We select a student at random from each school. Find the probability that the university A student's IQ is at least 5 points higher than the university B students IQ.

c) Select 3 university B students at random. Find the probability that this groups average IQ is at least 130 points.

3. The data in the table represents the ages of the winners of an award for the past five years. Use the data to answer questions.

37 42

41 45

46

a) Find the population mean age of the five winners

b) For a sample size of three construct a table of all possible samples and their sample means.

c) Draw a dot plot for the sampling of the sample mean for the sample size of three.

d) For a random sample size 3. What is the chance that the sample mean will equal the population mean.

e) For a random sample size 3, obtain the probability that the sampling error made in estimating the population mean by the sample mean will be 1 year or less, that is determine the probability that x be within 1 year of the mean.

4. Complete parts (a) through (e) for the population data below 3, 4, 5

a) Find the mean, m, of the variable

b) For each of the possible sample sizes construct a table with all possible samples and their sample means, and draw a dotplot for sampling distribution of the sample mean. The sample mean x is found by summing the observations in the sample and dividing by the sample size.

c) Use the dotplots from part (b) to create one plot with the sampling distributions for each sample size and interpret your results. The correct plot is shown below.

d) For each of the possible sample sizes find the probability that the sample mean will equal the population mean.

e) For each of the possible sample sizes find the probability that the sampling error made in estimating the population mean by the sample mean will be 0.5 or less (in magnitude) that is that the absolute value of the difference between the sample mean and the population mean is almost 0.5.

5. The data in the table represent the ages of the winners of an award for the past five years. Use the data to answer questions (a) through (c).

a) Determine the population mean age, M, of the five numbers.

b) Consider a sample size of 2 without replacement. Find the mean of the variable x

6. A variable of a population has a mean of m=73 and a standard deviation of x=7.

a) Identify the sampling distribution of the sample mean for samples of size 49.

b) In answering part (a) what assumptions did you make about the distribution of the variable?

c) Can you answer part (a) if the sample size is 25 instead of 49? Why or why not?

d) What is the shape of the sampling distribution? Uniform, skewed, normal, bimodal?

e) What is the mean of the sampling distribution?

f) What is the standard deviation of the sampling distribution?

7. According to one study brain weights of men are normally distributed with a mean of 1.60 kg and standard deviation of 0.12kg. Use the data to answer questions (a) through (e).

a) Determine the sampling distribution of the sample mean for samples of size 3. The sample mean is m-=

b) Determine the sampling distribution of the sample mean for samples of size 12.

c) Construct a graph of the normal population distribution and the two sampling distributions for brain weights.

d) Determine the percentage of all samples of three men that have mean brain weights within 0.1kg of the population mean brain weights of 1.70kg.

e) Determine the percentage of all samples of twelve men that have mean brain weights within 0.1kg the population mean brain weights of 1.70kg.

8. The mean annual salary of classroom teachers is approximately normal with a mean of $45,800. Assume a standard deviation of $8800. Do the following for the variable "annual salary" of classroom teachers.

a) Determine the sampling distribution of the sample mean for samples of size 64.

b) The sample mean is?

c) Determine the sampling of the sample mean for sample sizes 256

d) What is the probability that the sampling made in estimating the population mean salary of all classroom teachers by the mean salary of a sample of 64 classroom teachers will be at most $1000?

e) What is the probability that the sampling error made estimating the population mean salary of all classroom teachers by the mean salary of a sample of 256 classroom teachers will be at most $1000?