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Unit 11: Sampling Theory - Practice Quiz

ECAP790 60 Questions
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1 What is the main purpose of sampling theory?

Sampling Theory Easy
A. To eliminate all variation from collected data
B. To study every member of a population
C. To draw conclusions about a population from a sample
D. To arrange population values in ascending order

2 Which term describes a numerical value calculated from a sample?

Sampling Theory Easy
A. Population
B. Variable
C. Statistic
D. Parameter

3 In a simple random sample, how are the population members selected?

Random Samples and Random Numbers Easy
A. Only the largest members can be selected
B. The researcher selects the easiest members
C. Members are selected in alphabetical order
D. Each member has an equal chance of selection

4 What is a common use of random numbers in sampling?

Random Samples and Random Numbers Easy
A. Sorting observations from smallest to largest
B. Increasing the numerical values in the population
C. Replacing missing values with the sample mean
D. Selecting population members without personal preference

5 What happens after an item is selected in sampling with replacement?

Sampling with and without replacement Easy
A. It is replaced by the population mean
B. It is returned before the next selection
C. It is counted twice in the first selection
D. It is permanently removed from the population

6 Which statement is true about sampling without replacement?

Sampling with and without replacement Easy
A. The population size remains unchanged
B. Every selected item must be returned
C. A selected item cannot be selected again
D. All selections are always independent

7 What is a sampling distribution?

Sampling distributions Easy
A. The list of members excluded from sampling
B. The arrangement of one sample in a table
C. The distribution of a statistic from repeated samples
D. The distribution of values in one population

8 Which quantity can have a sampling distribution?

Sampling distributions Easy
A. Data source
B. Population label
C. Measurement unit
D. Sample mean

9 What is the mean of the sampling distribution of ?

Sampling distribution of means Easy
A.
B.
C.
D.

10 If the population standard deviation is and the sample size is , what is the standard error of ?

Sampling distribution of means Easy
A.
B.
C.
D.

11 What generally happens to the sampling distribution of the mean as the sample size increases?

Sampling distribution of means Easy
A. It becomes less spread out
B. It becomes more spread out
C. Its mean becomes undefined
D. Its center becomes zero

12 According to the central limit theorem, the sampling distribution of the mean tends to become what shape as increases?

Sampling distribution of means Easy
A. Approximately normal
B. Entirely irregular
C. Strongly rectangular
D. Perfectly uniform

13 If the population proportion is , what is the expected value of the sample proportion ?

Sampling distribution of properties Easy
A.
B.
C.
D.

14 What is the standard error of the sample proportion for independent observations?

Sampling distribution of properties Easy
A.
B.
C.
D.

15 For independent sample means and , what is the expected value of ?

Sampling distribution of differences and sums Easy
A.
B.
C.
D.

16 For sample means and , what is the expected value of ?

Sampling distribution of differences and sums Easy
A.
B.
C.
D.

17 What does a standard error measure?

Standard errors Easy
A. The largest error in a population
B. The total number of values in a sample
C. The difference between two data units
D. The variability of a statistic across samples

18 A smaller standard error usually indicates what about an estimate?

Standard errors Easy
A. It is always negative
B. It is always larger
C. It is more precise
D. It is more biased

19 In a software simulation of sampling, why are many random samples generated?

Software demonstration of elementary sampling theory Easy
A. To make every sample identical
B. To remove the population parameters
C. To approximate a sampling distribution
D. To avoid calculating any statistics

20 Which software output best displays the simulated sampling distribution of sample means?

Software demonstration of elementary sampling theory Easy
A. A list of population names
B. A table of measurement units
C. A paragraph describing the software
D. A histogram of the sample means

21 A college wants to estimate the average commuting time of its students. Which method is most likely to produce a representative sample?

Sampling Theory Medium
A. Survey the first 100 students entering the library
B. Survey 100 students who live in residence halls
C. Randomly select 100 students from the enrollment list
D. Randomly select 100 students from one academic department

22 A population contains individuals numbered 001 through 500. Three-digit random numbers are used to select a sample without replacement. How should the researcher handle the generated numbers?

Random Samples and Random Numbers Medium
A. Accept 000–500 and retain repeated valid numbers
B. Accept 001–500 and ignore only repeated valid numbers
C. Accept 001–500 and ignore 000, larger values, and repeats
D. Accept every number and divide values above 500 by two

23 A box contains 4 red and 6 blue balls. Two balls are drawn with replacement. What is the probability that both are red?

Sampling with and without replacement Medium
A.
B.
C.
D.

24 A population has and standard deviation . For a simple random sample of drawn without replacement, what is the approximate standard error of the sample mean?

Sampling with and without replacement Medium
A.
B.
C.
D.

25 Which statement best describes the sampling distribution of the sample mean for samples of size ?

Sampling distributions Medium
A. It is the distribution of population values below the mean
B. It is the distribution of sample sizes across repeated studies
C. It is the distribution of sample means from all possible samples
D. It is the distribution of all observations in one sample

26 A population has mean and standard deviation . For random samples of size , what are the mean and standard deviation of the sampling distribution of ?

Sampling distribution of means Medium
A. Mean and standard deviation
B. Mean and standard deviation
C. Mean and standard deviation
D. Mean and standard deviation

27 A strongly right-skewed population has a finite mean and variance. Which change most directly makes the sampling distribution of closer to normal?

Sampling distribution of means Medium
A. Increasing the population standard deviation
B. Decreasing the number of population members
C. Changing from a mean to a raw score
D. Increasing the random sample size

28 In a population, the proportion possessing a certain property is . For random samples of size , what are the approximate mean and standard deviation of the sample proportion ?

Sampling distribution of properties Medium
A. Mean and standard deviation
B. Mean and standard deviation
C. Mean and standard deviation
D. Mean and standard deviation

29 Two independent populations have standard deviations and . Samples of sizes and are taken. What is the standard error of ?

Sampling distribution of differences and sums Medium
A.
B.
C.
D.

30 Independent sample means have standard deviations and . What is the standard deviation of their sum?

Sampling distribution of differences and sums Medium
A.
B.
C.
D.

31 Two independent samples have , , , and . What is the approximate standard deviation of ?

Sampling distribution of differences and sums Medium
A.
B.
C.
D.

32 A sample of observations has sample standard deviation . What is the estimated standard error of the sample mean?

Standard errors Medium
A.
B.
C.
D.

33 The standard error of a sample mean is when the sample size is . Assuming the population standard deviation remains fixed, what sample size is needed to reduce the standard error to ?

Standard errors Medium
A.
B.
C.
D.

34 A population is 60% urban and 40% rural. In a proportional stratified sample of people, how many individuals should be selected from each group?

Sampling Theory Medium
A. 120 urban and 80 rural
B. 80 urban and 120 rural
C. 100 urban and 100 rural
D. 140 urban and 60 rural

35 A researcher uses a computer-generated random integer from 1 to 50 to select the first person, then chooses every 50th person from an ordered list. What sampling method is being used?

Random Samples and Random Numbers Medium
A. Systematic sampling
B. Cluster sampling
C. Stratified sampling
D. Convenience sampling

36 A population has 20 members, of whom 6 possess a particular property. If 5 members are sampled without replacement, which expression gives the probability that exactly 2 sampled members possess the property?

Sampling with and without replacement Medium
A.
B.
C.
D.

37 A population has a property proportion of . Ten independent selections are made with replacement. Which expression gives the probability of exactly two selections having the property?

Sampling with and without replacement Medium
A.
B.
C.
D.

38 A normally distributed population has mean and standard deviation . What is for a random sample of size ?

Sampling distribution of means Medium
A.
B.
C.
D.

39 Software repeatedly draws 5000 random samples of size from a population with mean . The average of the 5000 simulated sample means should be closest to which value?

Software demonstration of elementary sampling theory Medium
A.
B.
C.
D.

40 A student runs the same sampling simulation twice but obtains different results. Which software action would best make the random samples reproducible?

Software demonstration of elementary sampling theory Medium
A. Set the same random-number seed
B. Increase the number of histogram bins
C. Round every simulated value
D. Sort the population before sampling

41 A finite population has size and variance , where uses denominator . For a simple random sample without replacement of size , what is ?

Sampling Theory Hard
A.
B.
C.
D.

42 An unsigned 32-bit generator produces each integer with equal probability. Which procedure generates an exactly uniform integer from through ?

Random Samples and Random Numbers Hard
A. Return for every generated value
B. Accept only and return
C. Return without rejecting any generated value
D. Return the nearest integer to

43 Four ordered draws are made with replacement from a population of eight distinct units. What is the probability that exactly three distinct units appear?

Sampling with and without replacement Hard
A.
B.
C.
D.

44 Let be a random sample from , and let use denominator . Which statement is correct?

Sampling distributions Hard
A. and is independent of
B. and is correlated with
C. and is independent of
D. and is correlated with

45 Let be independent exponential variables with mean . Which expression equals ?

Sampling distribution of means Hard
A.
B.
C.
D.

46 A population contains units, of which have a specified property. An SRS without replacement of size is taken. What is the standard error of the sample proportion?

Sampling distribution of properties Hard
A.
B.
C.
D.

47 Independent SRSs without replacement are taken from two populations. Given and , what is ?

Sampling distribution of differences and sums Hard
A.
B.
C.
D.

48 For independent paired observations , suppose , , and . What is ?

Sampling distribution of differences and sums Hard
A.
B.
C.
D.

49 For paired observations, suppose , , , , and . Using the delta method, what is the approximate standard error of ?

Standard errors Hard
A.
B.
C.
D.

50 A simulation uses independent replications to estimate a tail probability and obtains . Ignoring uncertainty in the random-number generator, what is the Monte Carlo standard error of ?

Software demonstration of elementary sampling theory Hard
A.
B.
C.
D.

51 A one-stage cluster sample observes individuals in equal clusters of size . If the intracluster correlation is , what is the approximate effective sample size relative to an SRS?

Sampling Theory Hard
A.
B.
C.
D.

52 For a random sample of size from a continuous distribution with CDF , let be the third order statistic. What is the distribution of ?

Random Samples and Random Numbers Hard
A.
B.
C.
D.

53 In an SRS without replacement of size from , let and be inclusion indicators for two distinct population units. What is ?

Sampling with and without replacement Hard
A.
B.
C.
D.

54 In a multinomial sample of size , category probabilities are and . What is ?

Sampling distribution of properties Hard
A.
B.
C.
D.

55 A population has variance and third central moment . For an IID sample of size , what is the skewness of the sampling distribution of ?

Sampling distribution of means Hard
A.
B.
C.
D.

56 Let be IID with variance . Define and . What is ?

Sampling distribution of differences and sums Hard
A.
B.
C.
D.

57 Two independent samples have and . Which pair gives the Welch standard error of and the associated approximate degrees of freedom?

Standard errors Hard
A. ,
B. ,
C. ,
D. ,

58 Consider the Python simulation means = np.array([rng.choice(pop, n, replace=False).mean() for _ in range(B)]). Assuming pop is the complete finite population, what does means.std(ddof=1) estimate as becomes large?

Software demonstration of elementary sampling theory Hard
A. The average within-sample standard deviation under SRS without replacement
B. The Monte Carlo standard error of the average of all simulated means
C. The standard deviation of the finite population after bias correction
D. The standard error of the sample mean under SRS without replacement

59 In an unequal-probability design, sampled units have observed pairs and , where is the first-order inclusion probability. What is the Horvitz–Thompson estimate of the population total?

Sampling Theory Hard
A.
B.
C.
D.

60 For independent Bernoulli trials with , which is the most appropriate approximation to ?

Sampling distribution of properties Hard
A.
B.
C.
D.
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