Deck 8: Sampling Distributions

Question 1

What effect does increasing the sample size have on the sampling distribution of ?

A) The mean of the distribution gets smaller.
B) Sample size has no effect on the sampling distribution of .
C) The variability of the sampling distribution gets smaller.
D) The mean of the distribution gets larger.

Answer 1

The variability of the sampling distribution gets smaller.

Question 2

A department store claims that its average customer satisfaction rating is 28.5.We believe that this rating is too high, so we randomly sample 40 customers and find a sample average of 25.2.If the standard deviation of customer satisfaction for this store is 13.1, what is the probability of observing a sample at least this far below the claimed mean, assuming the claim is true?

A) 0.9441
B) 0.4013
C) 0.0559
D) 0.2500

Answer 2

0.0559

Question 3

In the natural world, the number of measurable variables that follow a normal distribution is quite remarkable.Many of these measurable variables are actually the sum of other independent variables.The Central Limit Theorem helps explain this phenomenon because:

A) the Central Limit Theorem states that all measurable variables follow a normal distribution.
B) the Central Limit Theorem states that for a sufficiently large number of variables, the sum of these variables will be approximately normally distributed.
C) the Central Limit Theorem states that for large sample sizes, all measurable variables will follow a normal distribution.
D) the Central Limit Theorem states that because nature is normative, variables found in nature will be at least approximately normal.

Answer 3

the Central Limit Theorem states that for a sufficiently large number of variables, the sum of these variables will be approximately normally distributed.

Question 4

A simple random sample (SRS) of size n is best defined as:

A) a sample selected in such a way that every possible sample of size n will be selected.
B) a sample selected in such a way that every possible sample of size n has the same chance of being selected.
C) a sample selected in such a way that every possible sample of size n has a randomly assigned chance of being selected.
D) a sample selected in such a way that every possible sample of size n has some chance of being selected.

Question 5

Suppose you are interested in determining the average word length in Lincoln's Gettysburg Address.You decide to estimate this average word length by using a sample of 10 randomly selected words.Which of the following methods should result in a simple random sample of 10 words from the Address?

A) Have a student select 10 "representative" words from a copy of the Address.
B) Obtain a copy of the Address on a page of paper.Close your eyes and point the tip of your pencil on the page five times, each time selecting the word that your pencil lands on or nearest.
C) Obtain a list of the 268 words in the Address.Number the words.Use a random number generator to determine which 10 words to select.
D) Select every 27th word in the Address.

Question 6

If the monthly rents of all one-bedroom apartments in a small town are known to be normally distributed with a mean of $175 and a standard deviation of $35, which of the following would be the least likely individual rent that you might expect to find?

A) $135
B) $140
C) $210
D) $225

Question 7

Let X be the fuel efficiency, in miles per gallon, of a particular vehicle. Suppose X ~ N(μ = 18.5, σ = 2.2).Find P(18.2 ≤ ≤ 19) if n = 5.

A) 0.6950
B) 0.9839
C) 0.1467
D) 0.3129

Question 8

The ______________ of a statistic is the probability distribution of the statistic.

A) limiting distribution
B) exponential distribution
C) normal distribution
D) sampling distribution

Question 9

Homeowners who recently bought their houses from a local developer allege that 30% of the houses this developer constructs have some major defect that will require substantial repairs.To test this allegation, we randomly sample 20 houses constructed by the developer and find that two of the homes do, indeed, have some major defect.If the allegation is correct, what is the probability of observing at most two defective homes out of a random sample of 20?

A) 0.0256
B) 0.1000
C) 0.0300
D) 0.9744

Question 10

An industrial fan is designed to move 2500 ft3of air per minute, on average.Under normal operating conditions, the standard deviation of air volume moved by this fan is 300 ft3/min.If the fan is operating as designed, what is the probability that the average of 30 randomly selected measurements of fan capacity will be less than 2000 ft3/min?

A) 0.0475
B) 0.0000
C) 0.9525
D) 0.913

Question 11

Consider the following discrete population distribution: ​ ​
We randomly draw samples of size 5 from this population using the simple random sampling technique.If we know that μx = 5.4, σx= 2.069, what are the mean and the standard deviation of the sampling distribution of sample means? Assume the Central Limit Theorem applies.

A) μ = 5.4, σ = 2.069
B) μ = 1.08, σ = 0.4138
C) μ = 5.4, σ = 0.9253
D) μ = 5.4, σ = 0.4138

Question 12

A psychologist randomly samples 28 clinically depressed patients.She subjects each patient to a standardized battery of tests.The average battery test score for the sample is 128.In this example, the number 128 is a:

A) parameter.
B) sampling distribution.
C) statistic.
D) random variable.

Question 13

A _________ is any quantity computed from values in a sample.

A) sampling distribution
B) parameter
C) statistic
D) parametric statistic

Question 14

Concerning the sampling distribution of , what is the significance of the conditions np ≥ 5 and n(1 - p) ≥ 5?

A) It is impossible to make any inference on p using unless both these conditions are met.
B) The sampling distribution of will be approximately normal if both these conditions are met.
C) It is possible to use to estimate p only if one of these conditions is met.
D) The value of will be very close to p if and only if the sample size is large enough for both these conditions to be met.

Question 15

An exponential population has parameter λ = 0.012 (hence, μ= 83.33, σ2= 6944.44).What is the probability that a random sample of 41 from this population will average between 81 and 84?

A) 0.5199
B) 0.0160
C) 0.0000
D) 0.0913

Question 16

If the sample size is large (n ≥ 30), then, regardless of the population distribution, which of the following summarizes the sampling distribution of random variable ?

A)
B)
C)
D)

Question 17

Because

A) will always be the same as p.
B) is unbiased for p.
C) must always be ≤ p.
D) will always be ≥ p. ​

Question 18

A heavily right-skewed population has a mean of 28 and a standard deviation of 2.What is the probability that a random sample of 5 from this population will average less than 27?

A) 0.1314
B) 0.3085
C) 0.6915
D) We cannot answer this question because the sample size is small and the population is non-normal; the distribution of cannot be assumed to be normal.

Question 19

It is believed that 15% of people who fly on commercial airliners are "very concerned" about the safety of the carrier they have chosen.If this statement is accurate, what is the probability that out of 150 randomly selected people who fly on commercial airliners, between 20 and 25 of them are "very concerned" about the safety of the carrier that they have chosen?

A) 0.1333
B) 0.0333
C) 0.4314
D) 0.7157

Question 20

If a population has a variance of 64 and we take a simple random sample of size 25, what will the standard deviation of the sampling distribution of be, assuming the Central Limit Theorem holds in this case?

A) 0.32
B) 0.625
C) 1.60
D) 3.125

Question 21

If a population proportion is 0.8 and a sample of size n = 100 is randomly selected from this population, what is the standard deviation of the proportion ?

A) 0.0258
B) 0.0355
C) 0.0400
D) 0.0538

Question 22

Heights of children entering kindergarten are normally distributed with a mean of 103 cm and a standard deviation of 1.27 cm.What is the probability that a random sample of 8 children will have an average height more than 103.5 cm?

A) 0.8665
B) 0.6517
C) 0.3483
D) 0.1335

Question 23

A population of sports teams consists of 26 individual teams.We wish to randomly sample 4 teams for observation.If each team has an equal likelihood of being selected, what is the probability that any given sample of size 4 will be selected?

A) 1/14,950
B) 4/14,950
C) 1/358,800
D) 4/358,800

Question 24

For after-hours emergencies, doctors often use a messaging/paging service to allow patients to contact them.If it is known that 30% of the patients who call the messaging service fail to leave a message for the doctor, what is the probability that out of 80 randomly sampled calls, more than 30 will fail to leave a message?

A) 0.0512
B) 0.9279
C) 0.3750
D) 0.0721

Question 25

In a recent study, it was reported that the proportion of employees who miss work on Fridays is 0.15, and that the standard deviation of the sample proportion is .However, the report did not indicate the sample size n.What was the sample size, assuming the Central Limit Theorem holds in this case?

A) 26
B) 58
C) 102
D) 204

Question 26

To better understand the student body, a college administrator takes a census of all students currently enrolled at the college.From the census, he finds that the range of ages at this school is 41 years.In this example, 41 years is a:

A) parameter.
B) statistic.
C) sampling distribution.
D) random variable.

Question 27

Which of the following statements is true about the sampling distribution of the sample proportion ?

A) The mean of the sampling distribution will equal the population proportion p.
B) An increase in the sample size n will result in a decrease in the standard error of .
C) The sampling distribution will be approximately normal, provided that np ≥ 5 and n(1-p) ≥ 5.
D) All of the above are true.

Question 28

Which of the following statements about the Central Limit Theorem is true?

A) The Central Limit Theorem states that the sample mean is always equal to the population mean μ.
B) The Central Limit Theorem states that the sampling distribution of the population mean μ is approximately normal provided that n ≥ 30.
C) The Central Limit Theorem states that the sampling distribution of the sample mean is approximately normal for large sample sizes (n ≥ 30).
D) The Central Limit Theorem states that the sample mean is equal to the population mean μ provided that n ≥ 30.

Question 29

A math professor once claimed that 90% of students pass a particular class of hers.To check this claim, a random sample of 150 students who had taken this class was obtained.It was found that 129 passed the class.If the professor is correct, what is the probability that 129 or fewer of the students in such a sample would have passed the class?

A) 0.9484
B) 0.0516
C) 0.5516
D) 0.4484

Question 30

We wish to record the results from two six-sided die rolls.How many unique outcomes are possible?

A) 15
B) 36
C) 30
D) 64

Question 31

In the normal process of simple random sampling (SRS), samples are almost always drawn without replacement.Why is this a problem for the assumptions involved with a true SRS?

A) Because the samples are drawn without replacement, there will never be enough samples to accurately calculate a meaningful statistic.
B) Because the samples are drawn without replacement, the process is not really random.
C) Because the samples are drawn without replacement, the samples are not truly independent, as the probability of one sample is affected by previous selections.
D) Because the samples are drawn without replacement, this creates complexity, which violates the assumption of simplicity.

Question 32

Consider a large population with a mean of 150 and a standard deviation of 27.A random sample of size 36 is taken from this population.What is the standard error of the sampling distribution of the sample mean?

A) 4.17
B) 4.50
C) 5.20
D) 5.56

Question 33

If the standard error of the sampling distribution of a sample proportion is 0.02049 for samples of size 500, then the population proportion must be either:

A) 0.2 or 0.8.
B) 0.3 or 0.7.
C) 0.6 or 0.4.
D) 0.15 or 0.85.

Question 34

Suppose we know that the proportion of objects in a population that have a particular characteristic of interest is 0.10.Are there any concerns if we want to calculate probabilities for the sampling distribution of n = 40 using a normal approximation?

A) Because nis large, our only concern with using a normal approximation will be the small amount of variability in the sampling distribution of .
B) Because is small here, the sampling distribution of should not be assumed to be approximately normal.
C) Because ≤ 0.5, the sampling distribution of should not be assumed to be approximately normal.
D) If n < 5, then a normal approximation is not appropriate.
​

Question 35

If the population proportion is 0.38, what is if we calculate using 20 samples?

A) 0.0118
B) 0.1085
C) 0.0190
D) Because is small, we cannot calculate accurately.

Question 36

In a major city in the northwest United States, 28% of the days experience some period of rain.What is the probability that more than 12 days out of a random sample of 31 days will experience some period of rain?

A) 0.0918
B) 0.0806
C) 0.9082
D) 0.3871

Question 37

The standard error of a statistic used as an estimator of a population parameter is:

A) the standard deviation of the sampling distribution of the statistic.
B) the variance of the sampling distribution of the statistic.
C) the same value as the population standard deviation.
D) the square root of the population variance.

Question 38

The survival times of adult mosquitoes exposed to a commonly used synthetic pesticide (Phenothrin) follow a gamma distribution, with a mean time until extermination of 48 sec and a standard deviation of 15.18 sec.What is the probability that a random sample of 40 mosquitoes will have an average survival time between 47.5 sec and 48 sec?

A) 0.0832
B) 0.4207
C) 0.4880
D) 0.0120

Question 39

In a study whose results were published in the British Medical Journal (Willis et al., 2004), researchers tested whether dogs can smell cancer in humans.Each dog used in the study was presented with seven urine samples to smell, with only one of the urine samples coming from a patient who had bladder cancer.Taken as a group, the dogs were tested on 54 trials and correctly identified the urine from the cancer patient in 22 trials.If the dogs were just guessing and so had a probability of success p = 1/7, what is the probability they would be correct in 22 or more of the 54 trials?

A) 0.000
B) 0.042
C) 0.087
D) 0.168

Question 40

The speed at which an automated assembly line produces a product follows a normal distribution, with a mean production time of 32.20 sec and a standard deviation of 1.05 sec.A full production run from this line consists of 20 completed products.What is the probability that a full production run will take less than 32 sec, on average, to finish?

A) 0.4247
B) 0.1977
C) 0.1900
D) 0.5753

Question 41

Suppose 45% of voters favor candidate A.We can be 67% sure that out of 200 randomly selected voters, at least how many will favor candidate A?

A) 90
B) 87
C) 134
D) 93

Question 42

If the population is perfectly normal, the sampling distribution of will be:

A) exactly normal for any sample size.
B) approximately normal for any sample size.
C) exactly normal only if the sample size is large enough.
D) approximately normal only if the sample size is large enough.

Question 43

If the sample size (n) is large (say, n ≥ 30), then it is safe to model the sampling distribution of as a:

A) discrete distribution.
B) continuous distribution.
C) log-normal distribution.
D) normal distribution.

Question 44

The problems associated with sampling without replacement in the simple random sample process are negligible provided:

A) the sample is large relative to the population size.
B) the population is small relative to the sample size.
C) the sample and the population are of roughly the same size.
D) the population is large relative to the sample size.

Question 45

We wish to survey high schools in a particular city.If the city has 22 high schools and we want to select 5 of them, how many unique samples of size 5 are possible?

A) 3,160,080
B) 5,153,632
C) 26,334
D) 2.384 × 1015

Question 46

A sample of 82 mice is observed from birth until they are one year old.The total amount of growth experienced by each mouse is recorded.The median amount of growth of these mice is 4.5 cm.In this example, 4.5 cm is a:

A) parameter.
B) sampling distribution.
C) statistic.
D) random variable.

Question 47

All statistics may be treated as:

A) characteristics of a population.
B) fixed quantities.
C) round variables.
D) random variables.

Question 48

If n ≥ 30, we know that the sampling distribution of is:

A) approximately normal.
B) exactly normal.
C) suitable for estimating the unknown parameter p.
D) unknown, because n ≥ 30 alone tells us nothing substantial concerning the sampling distribution of .

Question 49

According to the Central Limit Theorem, as the sample size (n) from a non-normal population increases:

A) the mean of the sampling distribution of becomes closer to μ, the population mean.
B) the sample standard deviation decreases.
C) the sampling distribution of increasingly approximates a normal distribution.
D) there is less of a problem with nonrepresentative sample bias.

Question 50

If a population is skewed, based on the Central Limit Theorem, what is a good rule of thumb for assuming the sampling distribution of is approximately normal?

A) np ≥ 5
B) n ≥ 30
C) μ ≥ 30
D) n ≤ 30