Deck 7: Sampling Methods and the Central Limit Theorem

Question 1

i. A simple random sample assumes that each item or person in the population has an equal chance of being included.
Ii) We can expect some difference between sample statistics and the corresponding population parameters. This difference is called the sampling error.
Iii) A sampling distribution of the means is a probability distribution consisting of a list of all possible sample means of a given sample size selected from a population and the probability of occurrence associated with each sample mean

A) (i), (ii) and (iii) are all correct statements
B) (i) and (ii) are correct statements, but not (iii).
C) (i) and (iii) are correct statements but not (ii).
D) (ii) and (iii) are correct statements but not (i).
E) All statements are false

Answer 1

A

Question 2

A province-wide sample survey is to be made. First, the province is subdivided into counties. Seven counties are selected at random and further sampling is concentrated on these seven counties. What type of sampling is this?

A) Simple random
B) Nonproportional
C) Cluster
D) Stratified
E) None of the choices are correct

Answer 2

C

Question 3

i. It is often not feasible to study the entire population because it is impossible to check all the items in the population.
ii. Sampling a population is often necessary because the cost of studying all the items in the population is prohibitive.
Iii) Sampling is a sign of laziness on the part of the statistician.

A) (i), (ii) and (iii) are all correct statements
B) (i) and (ii) are correct statements, but not (iii).
C) (i) and (iii) are correct statements but not (ii).
D) (ii) and (iii) are correct statements but not (i).
E) All statements are false

Answer 3

(i) and (ii) are correct statements, but not (iii).

Question 4

Suppose we select every fifth invoice in a file. What type of sampling is this?

A) Simple random
B) Stratified random
C) Cluster
D) Systematic
E) None of the choices are correct

Question 5

What is the difference between a sample mean and the population mean called?

A) Standard error of the mean
B) Sampling error
C) Interval estimate
D) Point estimate
E) None of the choices are correct

Question 6

What is it called when all the items in a population have a chance of being selected in a sample?

A) Random sampling
B) z-score
C) Sampling error
D) Nonprobability sampling
E) None of the choices are correct

Question 7

The mean of all possible sample means is equal to the

A) population variance.
B) ?²/n.
C) sample variance.
D) population mean.
E) None of the choices are correct.

Question 8

As the size of the sample increases, what happens to the shape of the sampling means?

A) Cannot be predicted in advance
B) Approaches a normal distribution
C) Positively skewed
D) Negatively skewed
E) None of the choices are correct

Question 9

i. If probability sampling is done, each item in the population has a chance of being chosen.
ii. If the size of a sample equals the size of the population, we would not expect any error in estimating the population parameter.
Iii) Ontario Parks wardens estimate the average weight of the fish or game population by using creel checks and other devices. Based on this sample data, a warden might estimate that the mean weight of Coho salmon caught in Lake Ontario is 2.5 pounds. This single number is called a point estimate of
The unknown population parameter.

A) (i), (ii) and (iii) are all correct statements
B) (i) and (ii) are correct statements, but not (iii).
C) (i) and (iii) are correct statements but not (ii).
D) (ii) and (iii) are correct statements but not (i).
E) All statements are false

Question 10

i. If the sample size keeps getting larger and larger and finally equals the size of the population, there would be no error in predicting the population mean because the sample size and the size of the population would be the same.
ii. A simple random sample assumes that each item or person in the population has an equal chance of being included.
Iii) We can expect some difference between sample statistics and the corresponding population parameters. This difference is called the sampling error.

A) (i), (ii) and (iii) are all correct statements
B) (i) and (ii) are correct statements, but not (iii).
C) (i) and (iii) are correct statements but not (ii).
D) (ii) and (iii) are correct statements but not (i).
E) All statements are false

Question 11

Which of the following is the standard error of the mean?

A) µ
B) x/n
C) $\bar{x}$
D) s/?n
E) None of the choices are correct

Question 12

A group of statistics students decided to conduct a survey at their university to find the average (mean) amount of time students spend studying per week. Based on a simple random sample, they surveyed
144 students. The statistics showed that students studied an average of 20 hours per week with a standard deviation of 10 hours.
What is the standard error of the mean?

A) 0.83
B) 10
C) 0.5
D) 2
E) None of the choices are correct

Question 13

An accounting firm is planning for the next tax preparation season. From last year's returns, the firm collects a systematic random sample of 100 filings. The 100 filings showed an average preparation time of 90 minutes with a standard deviation of 140 minutes.
What is the standard error of the mean?

A) 14 minutes
B) 140 minutes
C) 1.4 minutes
D) 90 minutes
E) None of the choices are correct

Question 14

Suppose we select every fifth invoice in a file. What type of sampling is this?

A) Random
B) Cluster
C) Stratified
D) Systematic
E) None of the choices are correct

Question 15

Manufacturers were subdivided into groups by volume of sales. Those with more than $100 million in sales were classified as Class A large; those from $50 to $100 million as Class A medium size; and those between $25 and $50 million..., and so on. Samples were then selected from each of these groups. What is this type of sampling called?

A) Simple random
B) Stratified random
C) Cluster
D) Systematic
E) None of the choices are correct

Question 16

i. An estimate of the population mean based on a large sample is less reliable than an estimate made using a small sample.
Ii) The standard error of the mean will vary according to the size of the sample that is in the denominator. As the sample size n gets larger, the variability of the sample means gets smaller.
Iii) To determine the value of the standard error of the mean, the total error is divided by the sample size.

A) (i), (ii) and (iii) are all correct statements
B) (i) is a correct statement, but not (ii) or (iii).
C) (ii) is a correct statement, but not (i) or (iii).
D) (ii) and (iii) are correct statements but not (i).
E) All statements are false

Question 17

An experiment involves randomly selecting a sample of 256 middle managers for study. One item of interest is their mean annual income. The sample mean is computed to be $35,420 and the sample standard deviation is $2,050. What is the standard error of the mean?

A) $5.65
B) $128.13
C) $138.36
D) $2,050
E) $8.01

Question 18

Sampling error is the difference between a corresponding sample statistic and the

A) sample mean.
B) biased sample.
C) population parameter.
D) chance error.
E) None of the choices are correct.

Question 19

All possible samples of size n are selected from a population and the mean of each sample is determined. What is the mean of the sample means?

A) Exactly the same as the population mean
B) Larger than the population mean
C) Smaller than the population mean
D) Cannot be estimated in advance
E) None of the choices are correct

Question 20

Manufacturers were subdivided into groups by volume of sales. Those with more than $100 million in sales were classified as Class A large; those from $50 to $100 million as Class A medium size; and those between $25 and $50 million..., and so on. Samples were then selected from each of these groups. What is this type of sampling called?

A) Simple random
B) Stratified random
C) Cluster
D) Systematic
E) None of the choices are correct

Question 21

Mileage tests were conducted on a randomly selected sample of 100 newly developed automobile tires. The average tread wear was found to be 50,000 miles with a standard deviation of 3,500 miles. What is the best estimate of the average tread life in miles for the entire population of these tires?

A) 50,000
B) 3,500
C) (50,000/100)
D) (3,500/100)
E) None of the choices are correct

Question 22

Suppose a research firm conducted a survey to determine the average amount of money steady smokers spend on cigarettes during a week. A sample of 100 steady smokers revealed that the sample mean is $20 and the sample standard deviation is $5. What is the probability that a sample of 100 steady smokers spend between $19 and $21?

A) 0.4772
B) 0.0228
C) 0.9544
D) $20
E) None of the choices are correct

Question 23

Suppose a research firm conducted a survey to determine the average amount of money steady smokers spend on cigarettes during a week. A sample of 100 steady smokers revealed that the sample mean is $80 and the sample standard deviation is $20. What is the probability that a sample of 100 steady smokers spend between $76 and $84?

A) 0.4772
B) 0.0228
C) 0.9544
D) 0.3400
E) 0.9999

Question 24

An accounting firm is planning for the next tax preparation season. From last year's returns, the firm collects a systematic random sample of 100 filings. The 100 filings showed an average preparation time of 90 minutes with a standard deviation of 140 minutes.
What is the probability of finding a sample mean less than 18 hours?

A) 0.4820
B) 0.4920
C) 0.0080
D) 0.0180
E) None of the choices are correct

Question 25

An accounting firm is planning for the next tax preparation season. From last year's returns, the firm collects a systematic random sample of 100 filings. The 100 filings showed an average preparation time of 90 minutes with a standard deviation of 140 minutes. What is the probability that the mean completion time is between 1 and 2 hours, i.e., 60 and 120 minutes?

A) Approximately 1.
B) 0.1664
C) 0.8336
D) 0.9676
E) None of the choices are correct

Question 26

An accounting firm is planning for the next tax preparation season. From last year's returns, the firm collects a systematic random sample of 100 filings. The 100 filings showed an average preparation time of 90 minutes with a standard deviation of 140 minutes.
What is the probability that the mean completion time will be more than 120 minutes?

A) Approximately zero
B) 0.0832
C) 0.4168
D) 0.0162
E) None of the choices are correct

Question 27

An accounting firm is planning for the next tax preparation season. From last year's returns, the firm collects a systematic random sample of 100 filings. The 100 filings showed an average preparation time of 90 minutes with a standard deviation of 140 minutes.
What is the probability that average student study time is between 18 and 22 hours?

A) 0.9640
B) 0.0160
C) 0.0360
D) 0.9840
E) None of the choices are correct

Question 28

A new extended-life light bulb has an average service life of 750 hours, with a standard deviation of 50 hours. The shape of this distribution is unknown. From a sample of 100 light bulbs, about what percent of the bulbs will last more than 700 hours?

A) 100%
B) 34.13%
C) 84.13%
D) 50%
E) None of the choices are correct

Question 29

An accounting firm is planning for the next tax preparation season. From last year's returns, the firm collects a systematic random sample of 100 filings. The 100 filings showed an average preparation time of 90 minutes with a standard deviation of 140 minutes.
What assumptions do you need to make about the shape of the population distribution of all possible tax preparation times to make inferences about the average time to complete a tax form?

A) The population distribution is skewed to the right.
B) The population distribution is skewed to the left.
C) The population distribution is normal.
D) The shape of the population distribution does not matter.
E) None of the choices are correct

Question 30

An accounting firm is planning for the next tax preparation season. From last year's returns, the firm collects a systematic random sample of 100 filings. The 100 filings showed an average preparation time of 90 minutes with a standard deviation of 140 minutes.
What is the probability that a sample mean would exceed 20 hours per week?

A) 1.0
B) 0.5
C) 1.96
D) Cannot be calculated based on the given information.
E) None of the choices are correct

Question 31

The Intelligence Quotient (IQ) test scores are normally distributed with a mean of 100 and a standard deviation of 15.
I) What is the probability that a person would score 130 or more on the test?
Ii) You enrolled in a class of 25 students. What is the probability that the class' average IQ exceeds
130?

A) 0.0228, 0.4987
B) 0.4772, 0.300
C) 0.4772, 0.0026
D) 0.0228, approximately zero
E) None of the choices are correct

Question 32

Prince Edward Island Fisheries and Environment has been feeding a special food to rainbow trout fingerlings in a pond. A sample of the weights of 40 trout revealed that the mean weight is 402.7 grams and the standard deviation 8.8 grams. What is the probability that the mean weight for a sample of 40 trout exceeds 405.5 grams?

A) 0.3783
B) 0.0228
C) 1.0
D) 0.5
E) None of the choices are correct

Question 33

An experiment involves selecting a random sample of 256 middle managers at random for study. One item of interest is their mean annual income. The sample mean is computed to be $35,420 and the sample standard deviation is $2,050. What is the standard error of the mean?

A) $128.125
B) $138.36
C) $2,050
D) $8.01
E) None of the choices are correct

Question 34

For a population that is not normally distributed, the distribution of the sample means will:

A) be negatively skewed.
B) approach the normal distribution.
C) be positively skewed.
D) take the same shape as the population.
E) None of the choices are correct.

Question 35

The daily sales of a small retail store in Toronto for the last 365 days are normally distributed with a mean of $2,050, and a standard deviation of $300. From a sample of 49 days, what is the probability of having a sample mean more than $2,000?

A) 0.3790
B) 0.8790
C) 0.121
D) Approximately zero
E) Approximately 100%

Question 36

The Intelligence Quotient (IQ) test scores are normally distributed with a mean of 100 and a standard deviation of 15. i. What is the probability that a person would score between 85 and 115?
Ii) Given a class with 25 students, what is the probability that the class' average IQ score is between 85 and 115?

A) 0.3414, 0.0228
B) 0.6826, 0.9772
C) 0.6826, Approximately zero
D) 1.0, 0.9544
E) None of the choices are correct

Question 37

The mean weight of trucks traveling on a particular section of I-475 is not known. A provincial highway inspector needs an estimate of the mean. He selects a random sample of 49 trucks passing the weighing station and finds the mean is 15.8 tonnes, with a standard deviation of the sample of 4.2 tonnes. What is probability that a truck will weigh less than 14.3 tonnes?

A) 0.0062
B) 0.3632
C) 0.1368
D) 0.4938
E) None of the choices are correct

Question 38

The daily sales of a small retail store in Toronto for the last 365 days are normally distributed with a mean of $2,050, and a standard deviation of $300.
I) What is the probability of daily sales exceeding $2,500?
Ii) From a sample of 49 days, what is the probability of having a sample mean less than $2,500?

A) 0.4332, 0.0668
B) 0.0668, 0.9332
C) 0.4332, approximately 100%
D) 0.0668, Approximately 0%
E) 0.0668, Approximately 100%

Question 39

i. If a population is not normally distributed, the sampling distribution of the sample means tends to approximate a normal distribution.
ii. The Central Limit Theorem states that if the sample size n is sufficiently large, the sampling distribution of the means will be approximately normal no matter whether the population is normally distributed, skewed, or uniform.
Iii) Based on the sampling distribution of the means and the central limit theorem, the sample mean can be used as a good estimator of the population mean, assuming that the size of the sample is sufficiently large.

A) (i), (ii) and (iii) are all correct statements
B) (i) is a correct statement, but not (ii) or (iii).
C) (ii) is a correct statement, but not (i) or (iii).
D) (ii) and (iii) are correct statements but not (i).
E) All statements are false

Question 40

What sample statistic is used to estimate a population parameter?

A) Parameter
B) Sampling error
C) Point estimate
D) Interval estimate
E) None of the choices are correct

Question 41

GreenFacts, a non-profit organization, reports that the average adult in Europe consumed 17 litres of alcohol last year. The standard deviation is 2.1 litres. If we randomly select 42 European adults, what is the probability that the sample mean is more than 17.8 litres?

A) 0.0068
B) 0.4873
C) 0.0058
D) 0.4863
E) 0.0680

Question 42

Dawson's Repair Service orders parts from an electronic company who advertises its parts o be no more than 2% defective. What is the probability that Bill Dawson finds 4 or more parts out of a sample of 50 to be defective?

A) Almost 100%
B) Less than one half of one percent
C) About 1%
D) Almost 50%
E) About 2%

Question 43

A retailer claims that 90% of its customers are "pleased" or "very pleased" with the customer service. In a survey of 300 customers taken last week, what is the probability that 84% or more will be "pleased" or "very pleased" with the service?

A) 0.1736
B) 0.5736
C) 0.3464
D) Almost 100%
E) Almost no chance

Question 44

Dawson's Repair Service orders parts from an electronic company who advertises its parts o be no more than 2% defective. What is the probability that Bill Dawson finds 2 or more parts out of a sample of 60 to be defective?

A) 4.85%
B) 0.49%
C) 15.62%
D) Almost 50%
E) 22.96%

Question 45

The mean rent for a two-bedroom apartment in the Greater Toronto Area is $2,000 per month. The distribution of the monthly costs does not follow the normal distribution. In fact, it is positively skewed. What is the probability of selecting a sample of 36 two-bedroom apartments and finding the mean to be under $1,500 per month? The standard deviation of the sample is $300.

A) 34.13%
B) 84.13%
C) 50%
D) 100%
E) None of the choices are correct

Question 46

70% of North American women have pierced ears. In a survey of 49 women, what is the probability that less than 30 had pierced ears?

A) 0.0901
B) 0.5901
C) 0.9099
D) 0.4099

Question 47

Dawson's Repair Service orders parts from an electronic company who advertises its parts o be no more than 2% defective. What is the probability that Bill Dawson finds 3 or more parts out of a sample of 50 to be defective?

A) Almost 100%
B) 0.0217
C) 15.62%
D) Almost 50%
E) 34.38%

Question 48

Alpha Corporation receives a shipment of flour every morning from their supplier. The flour is in 40 kg bags and Alpha will reject any shipment that is more than 4% underweight. The foreman samples
60 bags with each shipment and if the bags average more than 4% underweight, the whole shipment is returned to the supplier. Determine the value for the standard error of the proportion.

A) 0.0250
B) 0.0253
C) 0.00064
D) 0.0064
E) 0.08

Question 49

A convenience store estimates that 25% of its customers come in to buy milk. What is the probability that out of the next 200 customers, 60 or fewer will buy milk?

A) 0.4484
B) 0.9484
C) 0.0516
D) Almost no chance
E) 0.0505

Question 50

A study of a company's practice regarding the payment of invoices revealed that on the average an invoice was paid 20 days after it was received. The standard deviation equalled five days. A sample of
25 invoices is selected. Assuming that the distribution is normal, what percent of the sampled invoices were paid within 15 days of receipt?

A) 100%
B) 37.91%
C) 34.13%
D) 86.74%
E) None of the choices are correct

Question 51

The mean rent for a two-bedroom apartment in the Greater Toronto Area is $2,000 per month. The distribution of the monthly costs does not follow the normal distribution. In fact, it is positively skewed. What is the probability of selecting a sample of 36 two-bedroom apartments and finding the mean to be at least $1,500 per month? The standard deviation of the sample is $300.

A) 34.13%
B) 84.13%
C) 50%
D) 100%
E) None of the choices are correct

Question 52

Dawson's Repair Service orders parts from an electronic company who advertises its parts o be no more than 2% defective. What is the probability that Bill Dawson finds 3 or more parts out of a sample of 60 to be defective?

A) 4.85%
B) 0.485%
C) 15.62%
D) Almost 50%
E) 22.96%

Question 53

Dawson's Repair Service orders parts from an electronic company who advertises its parts o be no more than 2% defective. What is the probability that Bill Dawson finds 4 or more parts out of a sample of 50 to be defective?

A) Almost 100%
B) Less than one half of one percent
C) About 1%
D) Almost 50%

Question 54

The mean rent for a two-bedroom apartment in the Greater Toronto Area is $2,000 per month. The distribution of the monthly costs does not follow the normal distribution. In fact, it is positively skewed. What is the probability of selecting a sample of 36 two-bedroom apartments and finding the mean to be under $1,800 per month? The standard deviation of the sample is $300.

A) 34.13%
B) 0%
C) 50%
D) 100%
E) None of the choices are correct

Question 55

Dawson's Repair Service orders parts from an electronic company who advertises its parts o be no more than 2% defective. What is the probability that Bill Dawson finds 4 or more parts out of a sample of 60 to be defective?

A) 4.85%
B) 0.49%
C) 15.62%
D) Almost 50%
E) 22.96%

Question 56

A retailer claims that 90% of its customers are "pleased" or "very pleased" with the customer service. In a survey of 300 customers taken last week, what is the probability that between 80% and 90% will be "pleased" or "very pleased" with the service?

A) 0.1736
B) 0.5736
C) 0.3464
D) Almost 100%
E) Almost 50%

Question 57

Dawson's Repair Service orders parts from an electronic company who advertises its parts o be no more than 2% defective. What is the probability that Bill Dawson finds 2 or more parts out of a sample of 50 to be defective?

A) Almost 100%
B) 0.0217
C) 0.1562
D) Almost 50%
E) 0.4783

Question 58

70% of North American women have pierced ears. For a survey of 49 women, what is the standard error?

A) 0.5
B) 0.7
C) 0.0655
D) 0.4245
E) 0.3

Question 59

Alpha Corporation receives a shipment of flour every morning from their supplier. The flour is in 40 kg bags and Alpha will reject any shipment that is more than 5% underweight. The foreman samples
50 bags with each shipment and if the bags average more than 5% underweight, the whole shipment is returned to the supplier. Determine the value for the standard error of the proportion.

A) 0.03082
B) 0.0095
C) 0.00707
D) 0.00095
E) 0.09747

Question 60

It has been estimated that 25% of all university students switch majors within their first two years of starting classes. If a random sample of 500 third-year students is taken at a city university, what is an estimate of the probability that 20% or more had switched majors within their first two years?

A) .4951
B) .0049
C) .9951
D) 0.5059
E) None of these are correct

Question 61

What is the name of the standard deviation of the sampling distribution of the sample means called ____________.

Question 62

Alpha Corporation receives a shipment of flour every morning from their supplier. The flour is in 40 kg bags and Alpha will reject any shipment that is more than 5% underweight. The foreman samples
50 bags with each shipment and if the bags average more than 5% underweight, the whole shipment is returned to the supplier. Determine the value for the standard error of the proportion.

A) 0.03082
B) 0.00095
C) 0.00707
D) 0.09747
E) 0.0095

Question 63

For a sampling distribution of the means, what percent of the means would be between ± 1.96 standard deviations?

Question 64

Alpha Corporation receives a shipment of flour every morning from their supplier. The flour is in 40 kg bags and Alpha will reject any shipment that is more than 4% underweight. The foreman samples
60 bags with each shipment and if the bags average more than 4% underweight, the whole shipment is returned to the supplier. Determine the value for the standard error of the proportion.

A) 0.00132
B) 0.0253
C) 0.00064
D) 0.08898
E) 0.0250

Question 65

Alpha Corporation receives a shipment of flour every morning from their supplier. The flour is in 40 kg bags and Alpha will reject any shipment that is more than 5% underweight. The foreman samples
36 bags with each shipment and if the bags average more than 5% underweight, the whole shipment is returned to the supplier. Determine the value for the standard error of the proportion.

A) 0.00132
B) 0.0363
C) 0.00792
D) 0.08898
E) 0.1234

Question 66

Alpha Corporation receives a shipment of flour every morning from their supplier. The flour is in 40 kg bags and Alpha will reject any shipment that is more than 4% underweight. The foreman samples
40 bags with each shipment and if the bags average more than 4% underweight, the whole shipment is returned to the supplier. Determine the value for the standard error of the proportion.

A) 0.03098
B) 0.00096
C) 0.00064
D) 0.02771
E) 0.0250

Question 67

If the sampling size equals the population size, what is the sampling error?

Question 68

Which sampling method would be best to use if the population can be divided into homogeneous subgroups?

Question 69

What type of sampling is it when a population is first divided into subgroups and then a sample is selected from each subgroup?

Question 70

As the sample size (n) increases, does to the spread in the distribution of the sample means increase, decrease, or stay the same?

Question 71

Alpha Corporation receives a shipment of flour every morning from their supplier. The flour is in 40 kg bags and Alpha will reject any shipment that is more than 5% underweight. The foreman samples
60 bags with each shipment and if the bags average more than 5% underweight, the whole shipment is returned to the supplier. Determine the value for the standard error of the proportion.

A) 0.12264
B) 0.01504
C) 0.00792
D) 0.08898
E) 0.1234

Question 72

What shape will the sampling distribution of the mean be if a population is normally distributed?

Question 73

Is the standard deviation of the sampling distribution of the mean larger or smaller than the standard deviation of the population under study?

Question 74

For populations scattered in a wide area, what is the preferred technique for sampling?

Question 75

Which sampling method would you use if every k-th item in the population sequence is selected?

Question 76

Auditors may select every 20th file starting with say, the 5th file in the top drawer. Then file numbers 25, 45, 65, 85, . . . are audited. What type of sampling is this?_____ sampling.

Question 77

Is the mean of a population called a parameter or a statistic?

Question 78

Alpha Corporation receives a shipment of flour every morning from their supplier. The flour is in 40 kg bags and Alpha will reject any shipment that is more than 4% underweight. The foreman samples 50 bags with each shipment and if the bags average more than 4% underweight, the whole shipment is returned to the supplier. Determine the value for the standard error of the proportion.

A) 0.00132
B) 0.000768
C) 0.00064
D) 0.02771
E) 0.0250