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Deck 8: Sampling Variability and Sampling Distributions
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Question
For which of the following combinations of sample size and population proportion would the standard deviation of 
be smallest?
A)n = 40 and p = 0.3
B)n = 60 and p = 0.4
C)n = 100 and p = 0.5
D)n = 200 and p = 0.6
E)n = 300 and p = 0.7
be smallest? A)n = 40 and p = 0.3
B)n = 60 and p = 0.4
C)n = 100 and p = 0.5
D)n = 200 and p = 0.6
E)n = 300 and p = 0.7
Question
The closer p is to 0 or 1, the larger n must be in order for the distribution of 
to be approximately normal.
to be approximately normal. Question
Question
The principal at John F. Kennedy High School has been asked to provide the average number of classes taken by the students at KHS. Since the computer system is down, she takes her alphabetized list of students, randomly selects 50 students, determines the number of classes each of the 50 selected students is taking, and calculates 
= 5.4. She then reports to the PTA "Since I took a large random sample, the population mean number of classes taken by the students at KHS is 5.4." Write a short paragraph to send to her that explains why her statement is not correct.
= 5.4. She then reports to the PTA "Since I took a large random sample, the population mean number of classes taken by the students at KHS is 5.4." Write a short paragraph to send to her that explains why her statement is not correct. Question
Suppose that a particular candidate for public office is favored by 43% of all registered voters in the district A polling organization will take a random sample of 50 of these voters and will use 
, the sample proportion, to estimate p.
Determine the standard deviation of
.
A)0.005
B)0.035
C)0.100
D)0.043
E)0.070
, the sample proportion, to estimate p. Determine the standard deviation of
.
A)0.005
B)0.035
C)0.100
D)0.043
E)0.070
Question
Question
A sampling distribution of 
can be regarded as approximately normal when what conditions are met?
A)n ≥ 30
B)n ≥ 10
C)np ≥ 5 and n(1 - p) ≥ 5
D)n ≥ 5
E)np ≥ 10 and n(1 - p) ≥ 10
can be regarded as approximately normal when what conditions are met? A)n ≥ 30
B)n ≥ 10
C)np ≥ 5 and n(1 - p) ≥ 5
D)n ≥ 5
E)np ≥ 10 and n(1 - p) ≥ 10
Question
The distribution of 
will always have the same shape as the distribution of the population being sampled.
will always have the same shape as the distribution of the population being sampled. Question
A random sample is to be selected from a population that has a proportion of successes p = 0.50. Determine the mean and standard deviation of the sampling distribution of 
for the sample size n = 100.
A)mean:
= 0.50; standard deviation: 
= 0.05
B)mean:
= 0.50; standard deviation: 
= 0.075
C)mean:
= 0.50; standard deviation: 
= 0.041
D)mean:
= 0.85; standard deviation: 
= 0.48
E)mean:
= 0.85; standard deviation: 
= 0.048
for the sample size n = 100. A)mean:
= 0.50; standard deviation: = 0.05B)mean:
= 0.50; standard deviation: = 0.075C)mean:
= 0.50; standard deviation: = 0.041D)mean:
= 0.85; standard deviation: = 0.48E)mean:
= 0.85; standard deviation: = 0.048 Question
In a random sample of 79 students at a university, 7 had the flu.
Suppose you are interested in learning about the value of p, the proportion of all students at this university who have the flu. This proportion can be estimated using the sample proportion,
. What is the value of 
for this sample?
A)0.097
B)0.250
C)0.089
D)0.081
E)0.700
Suppose you are interested in learning about the value of p, the proportion of all students at this university who have the flu. This proportion can be estimated using the sample proportion,
. What is the value of for this sample?
A)0.097
B)0.250
C)0.089
D)0.081
E)0.700
Question
For n sufficiently large, the distribution of 
is approximately a standard normal distribution.
is approximately a standard normal distribution. Question
The sampling distribution of 
tends to be more spread out for larger sample sizes than for smaller sample sizes.
tends to be more spread out for larger sample sizes than for smaller sample sizes. Question
Question
The standard deviation of the distribution of 
decreases as n increases.
decreases as n increases. Question
The mean of the sampling distribution of 
is p no matter how large n is.
is p no matter how large n is. Question
Some colleges now allow students to rent textbooks for a semester. Suppose that 73% of all students enrolled at a particular college would rent textbooks if that option were available to them. If the campus bookstore uses a random sample of size 40 to estimate the proportion of students at the college who would rent textbooks, then what is the standard deviation of the sampling distribution of 
?
A)0.079
B)0.177
C)0.070
D)0.031
E)0.197
? A)0.079
B)0.177
C)0.070
D)0.031
E)0.197
Question
Question
Suppose that 8% of the customers of a cable television company watch the Jewelry Network channel at least once a week. The cable company does not know the actual proportion of all customers who watch the channel at least once a week and is trying to decide whether to replace this channel with a new local station. The company plans to take a random sample of 150 customers and to use 
as an estimate of the population proportion.
Determine the standard deviation of
.
A)0.022
B)0.006
C)2.400
D)0.070
E)0.041
as an estimate of the population proportion. Determine the standard deviation of
.
A)0.022
B)0.006
C)2.400
D)0.070
E)0.041
Question
As n grows larger, the mean of the sampling distribution of 
gets closer to μ.
gets closer to μ. Question
Which of the following symbols is used to indicate a population proportion?
A)p
B)
C)μ
D)σ
E)
A)p
B)
C)μ
D)σ
E)
Question
One method for estimating abundance of animals is known as line-intercept sampling. The theory of this method, when applied to Alaskan wolverines, predicts that π = 0.453 of attempts to locate wolverine tracks should be successful. Suppose that biologists will make 100 attempts to locate wolverine tracks in random locations in Alaska.
a)Show that this sample size is large enough to justify using the normal approximation to the sampling distribution of
.
b)What is the mean of the sampling distribution of
?
c)What is the standard deviation of the sampling distribution of
?
a)Show that this sample size is large enough to justify using the normal approximation to the sampling distribution of
.b)What is the mean of the sampling distribution of
?c)What is the standard deviation of the sampling distribution of
? Question
Consider sampling from a population whose proportion of successes is p = 0.5. As the sample size, n, increases, some characteristics of the sampling distribution of 
change. Which of the following characteristics will change as n increases, and what is the nature of the change?
a)The mean of the sampling distribution of
b)The standard deviation of the sampling distribution of 
c)The shape of the sampling distribution of 
change. Which of the following characteristics will change as n increases, and what is the nature of the change?a)The mean of the sampling distribution of
b)The standard deviation of the sampling distribution of c)The shape of the sampling distribution of Question
Determine the mean and standard deviation of the sampling distribution of 
when n = 100 and p = 0.65.
when n = 100 and p = 0.65. Question
Consider the following "population": {2, 2, 4, 5}. Suppose that a random sample of size n = 2 is to be selected without replacement from this population. There are 6 possible samples (since the order of selection does not matter). Compute the sample mean for each of these samples and use that information to construct the sampling distribution of 
. (Display it in table form.)
. (Display it in table form.) Question
The first large-scale study of the human sex ratio involved over 6,000 families each having 12 children. (This was done in 19th Century Germany--large families were more common then and there.) 52% of the children they observed were boys. Suppose that 21st Century researchers wish to replicate this observational study. In order to see if this proportion of boys might have changed in the intervening years, suppose the researchers track down 50 families with 12 children. From these 600 children, a random sample of 50 children is taken. 30 of the 50 children were boys ( 
= 0.6.)a)Show that it is reasonable to approximate the sampling distribution of 
using a normal distribution.
b)If the modern true population proportion of newborn boys is p = 0.52, what is the probability of observing a sample proportion of at least
= 0.6?
= 0.6.)a)Show that it is reasonable to approximate the sampling distribution of using a normal distribution.b)If the modern true population proportion of newborn boys is p = 0.52, what is the probability of observing a sample proportion of at least
= 0.6? Question
Consider sampling from a population whose proportion of successes is p = 0.1. As the sample size, n, increases, some characteristics of the sampling distribution of 
change. Which of the following characteristics will change as n increases, and what is the nature of the change?
a)The mean of the sampling distribution of
b)The standard deviation of the sampling distribution of 
c)The shape of the sampling distribution of 
change. Which of the following characteristics will change as n increases, and what is the nature of the change?a)The mean of the sampling distribution of
b)The standard deviation of the sampling distribution of c)The shape of the sampling distribution of Question
Question
One method for estimating the availability of office space in large cities is to conduct a random sample of offices, and calculate the proportion of offices currently being used. Suppose that real estate agents believe that p = 0.70 of all offices are currently occupied, and decide to take a sample to assess their belief. They are considering a sample size of n = 40.
a)Show that this sample size is large enough to justify using the normal approximation to the sampling distribution of
.
b)What is the mean of the sampling distribution of
if the real estate agents are correct?
c)What is the standard deviation of the sampling distribution of
if the real estate agents are correct?
d)If the real estate agents are correct, what is the probability that a sample proportion,
, would differ from p = 0.70 by as much as 0.05?
a)Show that this sample size is large enough to justify using the normal approximation to the sampling distribution of
.b)What is the mean of the sampling distribution of
if the real estate agents are correct?c)What is the standard deviation of the sampling distribution of
if the real estate agents are correct?d)If the real estate agents are correct, what is the probability that a sample proportion,
, would differ from p = 0.70 by as much as 0.05? Question
Consider the following "population": {1, 1, 3, 4}. Suppose that a random sample of size n = 2 is to be selected without replacement from this population. There are 6 possible samples (since the order of selection does not matter). Compute the sample mean for each of these samples and use that information to construct the sampling distribution of 
. (Display it in table form.)
. (Display it in table form.) Question
Consider sampling from a skewed population. As the sample size, n, increases, some characteristics of the sampling distribution of 
change. Does an increasing sample size cause changes in the characteristics of the sampling distribution shown below? If so, specifically how does the sampling distribution change?
a)The mean of the sampling distribution of
b)The standard deviation of the sampling distribution of 
c)The shape of the sampling distribution of 
change. Does an increasing sample size cause changes in the characteristics of the sampling distribution shown below? If so, specifically how does the sampling distribution change?a)The mean of the sampling distribution of
b)The standard deviation of the sampling distribution of c)The shape of the sampling distribution of Question
One method for estimating abundance of animals is known as line-intercept sampling. The theory of this method, when applied to Alaskan wolverines, predicts that p = 0.453 of attempts to locate wolverine tracks should be successful. Suppose that biologists will make 100 attempts to locate wolverine tracks in random locations in Alaska.
a)Show that this sample size is large enough to justify using the normal approximation to the sampling distribution of
.
b)What is the mean of the sampling distribution of
?
c)What is the standard deviation of the sampling distribution of
?
d)What is the probability that a sample proportion,
, would differ from p = 0.453 by as much as 0.05?
a)Show that this sample size is large enough to justify using the normal approximation to the sampling distribution of
.b)What is the mean of the sampling distribution of
?c)What is the standard deviation of the sampling distribution of
?d)What is the probability that a sample proportion,
, would differ from p = 0.453 by as much as 0.05? Question
Some biologists believe the evolution of handedness is linked to complex behaviors such as tool-use. Under this theory, handedness would be genetically passed on from parents to children. That is, left-handed parents would be more likely to have left-handed children than right-handed parents. An alternate theory asserts that handedness should be random, with left- and right-handedness equally likely. In a recent study using a simple random sample of n = 76 right-handed parents, 50 of the children born were right-handed. ( 
= 0.658.) Suppose handedness is a random occurrence with either hand equally likely to be dominant, implying that the probability of a right-handed offspring is p = 0.50.
a)Show that it is reasonable to approximate the sampling distribution of p using a normal distribution.
b)Assuming left- and right-handed children are equally likely from right-handed parents, what is the probability of observing a sample proportion of at least
= 0.658?
= 0.658.) Suppose handedness is a random occurrence with either hand equally likely to be dominant, implying that the probability of a right-handed offspring is p = 0.50.a)Show that it is reasonable to approximate the sampling distribution of p using a normal distribution.
b)Assuming left- and right-handed children are equally likely from right-handed parents, what is the probability of observing a sample proportion of at least
= 0.658? Question
Suppose that the actual proportion of people at a particular company who use their bicycles to travel to job is 0.35. In a study of bicycle parking places needs, administration would like to estimate this proportion. They plan to take a random sample of 50 employees and use the sample proportion who use bicycles, 
, as an estimate of the population proportion. Estimate the standard deviation of the sample distribution. Suppose that another sample of different size has been selected, and the standard deviation of this new sample is 
. Would you expect more or less sample-to-sample variability in the sample proportions than for when 
? Is the sample size that resulted in 
larger or smaller than 50?
A)The standard deviation for the initial sample of the size
is 
.Since the standard deviation 
is greater than 
, sample-to-sample variability is less and the new sample size is smaller.
B)The standard deviation for the initial sample of the size
is 
.Since the standard deviation 
is less than 
, sample-to-sample variability is more and the new sample size is larger.
C)The standard deviation for the initial sample of the size
is 
.Since the standard deviation 
is greater than 
, sample-to-sample variability is more and the new sample size is smaller.
D)The standard deviation for the initial sample of the size
is 
.Since the standard deviation 
is greater than 
, sample-to-sample variability is more and the new sample size is smaller.
E)The standard deviation for the initial sample of the size
is 
.Since the standard deviation 
is greater than 
, sample-to-sample variability is less and the new sample size is larger.
, as an estimate of the population proportion. Estimate the standard deviation of the sample distribution. Suppose that another sample of different size has been selected, and the standard deviation of this new sample is . Would you expect more or less sample-to-sample variability in the sample proportions than for when ? Is the sample size that resulted in larger or smaller than 50?A)The standard deviation for the initial sample of the size
is .Since the standard deviation is greater than , sample-to-sample variability is less and the new sample size is smaller.B)The standard deviation for the initial sample of the size
is .Since the standard deviation is less than , sample-to-sample variability is more and the new sample size is larger.C)The standard deviation for the initial sample of the size
is .Since the standard deviation is greater than , sample-to-sample variability is more and the new sample size is smaller.D)The standard deviation for the initial sample of the size
is .Since the standard deviation is greater than , sample-to-sample variability is more and the new sample size is smaller.E)The standard deviation for the initial sample of the size
is .Since the standard deviation is greater than , sample-to-sample variability is less and the new sample size is larger. Question
How are the quantities, p and 
, related?
, related? Question
The principal at Thomas Jefferson High School has been asked to estimate the proportion of students at KHS who drive to school and use the school parking lot. He takes a random sample of size n = 32 students and calculates a sample proportion, 
= 0.8. "Now," he exclaims, "since my sample size is greater than 30, the sampling distribution of the sample proportion is approximately normal." Write a short paragraph that explains why his statement is not correct.
= 0.8. "Now," he exclaims, "since my sample size is greater than 30, the sampling distribution of the sample proportion is approximately normal." Write a short paragraph that explains why his statement is not correct.
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Deck 8: Sampling Variability and Sampling Distributions
1
For which of the following combinations of sample size and population proportion would the standard deviation of be smallest?
A)n = 40 and p = 0.3
B)n = 60 and p = 0.4
C)n = 100 and p = 0.5
D)n = 200 and p = 0.6
E)n = 300 and p = 0.7
be smallest? A)n = 40 and p = 0.3
B)n = 60 and p = 0.4
C)n = 100 and p = 0.5
D)n = 200 and p = 0.6
E)n = 300 and p = 0.7
n = 300 and p = 0.7
2
The closer p is to 0 or 1, the larger n must be in order for the distribution of to be approximately normal.
to be approximately normal.True
3
What is a sampling distribution of a statistic?
The distribution of the possible values of a statistic is called its sampling distribution.
Note: Generally if the student communicates the idea that the sampling distribution associates different values of the statistic with their probabilities, he or she should get credit.
Note: Generally if the student communicates the idea that the sampling distribution associates different values of the statistic with their probabilities, he or she should get credit.
4
The principal at John F. Kennedy High School has been asked to provide the average number of classes taken by the students at KHS. Since the computer system is down, she takes her alphabetized list of students, randomly selects 50 students, determines the number of classes each of the 50 selected students is taking, and calculates = 5.4. She then reports to the PTA "Since I took a large random sample, the population mean number of classes taken by the students at KHS is 5.4." Write a short paragraph to send to her that explains why her statement is not correct.
= 5.4. She then reports to the PTA "Since I took a large random sample, the population mean number of classes taken by the students at KHS is 5.4." Write a short paragraph to send to her that explains why her statement is not correct. Unlock Deck
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5
Suppose that a particular candidate for public office is favored by 43% of all registered voters in the district A polling organization will take a random sample of 50 of these voters and will use , the sample proportion, to estimate p.
Determine the standard deviation of .
A)0.005
B)0.035
C)0.100
D)0.043
E)0.070
, the sample proportion, to estimate p. Determine the standard deviation of
.
A)0.005
B)0.035
C)0.100
D)0.043
E)0.070
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6
Which of the following statements is a population proportion?
A)A state census reports that 54% of the state's citizens owns at least one dog..
B)Out of 50 randomly-selected individuals, 19 indicated that they snore during sleep.
C)The mean lifetime of all cell phone batteries is 11.3 hours.
D)For a randomly-selected group of 20 computers running the Ubuntu operating system, the mean boot time was 28 seconds.
E)None of these
A)A state census reports that 54% of the state's citizens owns at least one dog..
B)Out of 50 randomly-selected individuals, 19 indicated that they snore during sleep.
C)The mean lifetime of all cell phone batteries is 11.3 hours.
D)For a randomly-selected group of 20 computers running the Ubuntu operating system, the mean boot time was 28 seconds.
E)None of these
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7
A sampling distribution of can be regarded as approximately normal when what conditions are met?
A)n ≥ 30
B)n ≥ 10
C)np ≥ 5 and n(1 - p) ≥ 5
D)n ≥ 5
E)np ≥ 10 and n(1 - p) ≥ 10
can be regarded as approximately normal when what conditions are met? A)n ≥ 30
B)n ≥ 10
C)np ≥ 5 and n(1 - p) ≥ 5
D)n ≥ 5
E)np ≥ 10 and n(1 - p) ≥ 10
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8
The distribution of will always have the same shape as the distribution of the population being sampled.
will always have the same shape as the distribution of the population being sampled. Unlock Deck
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9
A random sample is to be selected from a population that has a proportion of successes p = 0.50. Determine the mean and standard deviation of the sampling distribution of for the sample size n = 100.
A)mean: = 0.50; standard deviation: = 0.05
B)mean: = 0.50; standard deviation: = 0.075
C)mean: = 0.50; standard deviation: = 0.041
D)mean: = 0.85; standard deviation: = 0.48
E)mean: = 0.85; standard deviation: = 0.048
for the sample size n = 100. A)mean:
= 0.50; standard deviation: = 0.05B)mean:
= 0.50; standard deviation: = 0.075C)mean:
= 0.50; standard deviation: = 0.041D)mean:
= 0.85; standard deviation: = 0.48E)mean:
= 0.85; standard deviation: = 0.048 Unlock Deck
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10
In a random sample of 79 students at a university, 7 had the flu.
Suppose you are interested in learning about the value of p, the proportion of all students at this university who have the flu. This proportion can be estimated using the sample proportion, . What is the value of for this sample?
A)0.097
B)0.250
C)0.089
D)0.081
E)0.700
Suppose you are interested in learning about the value of p, the proportion of all students at this university who have the flu. This proportion can be estimated using the sample proportion,
. What is the value of for this sample?
A)0.097
B)0.250
C)0.089
D)0.081
E)0.700
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11
For n sufficiently large, the distribution of is approximately a standard normal distribution.
is approximately a standard normal distribution. Unlock Deck
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12
The sampling distribution of tends to be more spread out for larger sample sizes than for smaller sample sizes.
tends to be more spread out for larger sample sizes than for smaller sample sizes. Unlock Deck
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13
If p = 0.90, a sample size of n = 10 is large enough for the sampling distribution to be well approximated by a normal distribution.
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14
The standard deviation of the distribution of decreases as n increases.
decreases as n increases. Unlock Deck
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15
The mean of the sampling distribution of is p no matter how large n is.
is p no matter how large n is. Unlock Deck
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16
Some colleges now allow students to rent textbooks for a semester. Suppose that 73% of all students enrolled at a particular college would rent textbooks if that option were available to them. If the campus bookstore uses a random sample of size 40 to estimate the proportion of students at the college who would rent textbooks, then what is the standard deviation of the sampling distribution of ?
A)0.079
B)0.177
C)0.070
D)0.031
E)0.197
? A)0.079
B)0.177
C)0.070
D)0.031
E)0.197
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17
A statistic is a characteristic of the population.
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18
Suppose that 8% of the customers of a cable television company watch the Jewelry Network channel at least once a week. The cable company does not know the actual proportion of all customers who watch the channel at least once a week and is trying to decide whether to replace this channel with a new local station. The company plans to take a random sample of 150 customers and to use as an estimate of the population proportion.
Determine the standard deviation of .
A)0.022
B)0.006
C)2.400
D)0.070
E)0.041
as an estimate of the population proportion. Determine the standard deviation of
.
A)0.022
B)0.006
C)2.400
D)0.070
E)0.041
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19
As n grows larger, the mean of the sampling distribution of gets closer to μ.
gets closer to μ. Unlock Deck
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20
Which of the following symbols is used to indicate a population proportion?
A)p
B)
C)μ
D)σ
E)
A)p
B)
C)μ
D)σ
E)
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21
One method for estimating abundance of animals is known as line-intercept sampling. The theory of this method, when applied to Alaskan wolverines, predicts that π = 0.453 of attempts to locate wolverine tracks should be successful. Suppose that biologists will make 100 attempts to locate wolverine tracks in random locations in Alaska.
a)Show that this sample size is large enough to justify using the normal approximation to the sampling distribution of .
b)What is the mean of the sampling distribution of ?
c)What is the standard deviation of the sampling distribution of ?
a)Show that this sample size is large enough to justify using the normal approximation to the sampling distribution of
.b)What is the mean of the sampling distribution of
?c)What is the standard deviation of the sampling distribution of
? Unlock Deck
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22
Consider sampling from a population whose proportion of successes is p = 0.5. As the sample size, n, increases, some characteristics of the sampling distribution of change. Which of the following characteristics will change as n increases, and what is the nature of the change?
a)The mean of the sampling distribution of b)The standard deviation of the sampling distribution of c)The shape of the sampling distribution of
change. Which of the following characteristics will change as n increases, and what is the nature of the change?a)The mean of the sampling distribution of
b)The standard deviation of the sampling distribution of c)The shape of the sampling distribution of Unlock Deck
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23
Determine the mean and standard deviation of the sampling distribution of when n = 100 and p = 0.65.
when n = 100 and p = 0.65. Unlock Deck
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24
Consider the following "population": {2, 2, 4, 5}. Suppose that a random sample of size n = 2 is to be selected without replacement from this population. There are 6 possible samples (since the order of selection does not matter). Compute the sample mean for each of these samples and use that information to construct the sampling distribution of . (Display it in table form.)
. (Display it in table form.) Unlock Deck
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25
The first large-scale study of the human sex ratio involved over 6,000 families each having 12 children. (This was done in 19th Century Germany--large families were more common then and there.) 52% of the children they observed were boys. Suppose that 21st Century researchers wish to replicate this observational study. In order to see if this proportion of boys might have changed in the intervening years, suppose the researchers track down 50 families with 12 children. From these 600 children, a random sample of 50 children is taken. 30 of the 50 children were boys ( = 0.6.)a)Show that it is reasonable to approximate the sampling distribution of using a normal distribution.
b)If the modern true population proportion of newborn boys is p = 0.52, what is the probability of observing a sample proportion of at least = 0.6?
= 0.6.)a)Show that it is reasonable to approximate the sampling distribution of using a normal distribution.b)If the modern true population proportion of newborn boys is p = 0.52, what is the probability of observing a sample proportion of at least
= 0.6? Unlock Deck
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26
Consider sampling from a population whose proportion of successes is p = 0.1. As the sample size, n, increases, some characteristics of the sampling distribution of change. Which of the following characteristics will change as n increases, and what is the nature of the change?
a)The mean of the sampling distribution of b)The standard deviation of the sampling distribution of c)The shape of the sampling distribution of
change. Which of the following characteristics will change as n increases, and what is the nature of the change?a)The mean of the sampling distribution of
b)The standard deviation of the sampling distribution of c)The shape of the sampling distribution of Unlock Deck
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27
When the sample size is "large enough," the statistic p has an approximately normal sampling distribution. How does one determine if a sample size is large enough?
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28
One method for estimating the availability of office space in large cities is to conduct a random sample of offices, and calculate the proportion of offices currently being used. Suppose that real estate agents believe that p = 0.70 of all offices are currently occupied, and decide to take a sample to assess their belief. They are considering a sample size of n = 40.
a)Show that this sample size is large enough to justify using the normal approximation to the sampling distribution of .
b)What is the mean of the sampling distribution of if the real estate agents are correct?
c)What is the standard deviation of the sampling distribution of if the real estate agents are correct?
d)If the real estate agents are correct, what is the probability that a sample proportion, , would differ from p = 0.70 by as much as 0.05?
a)Show that this sample size is large enough to justify using the normal approximation to the sampling distribution of
.b)What is the mean of the sampling distribution of
if the real estate agents are correct?c)What is the standard deviation of the sampling distribution of
if the real estate agents are correct?d)If the real estate agents are correct, what is the probability that a sample proportion,
, would differ from p = 0.70 by as much as 0.05? Unlock Deck
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29
Consider the following "population": {1, 1, 3, 4}. Suppose that a random sample of size n = 2 is to be selected without replacement from this population. There are 6 possible samples (since the order of selection does not matter). Compute the sample mean for each of these samples and use that information to construct the sampling distribution of . (Display it in table form.)
. (Display it in table form.) Unlock Deck
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30
Consider sampling from a skewed population. As the sample size, n, increases, some characteristics of the sampling distribution of change. Does an increasing sample size cause changes in the characteristics of the sampling distribution shown below? If so, specifically how does the sampling distribution change?
a)The mean of the sampling distribution of b)The standard deviation of the sampling distribution of c)The shape of the sampling distribution of
change. Does an increasing sample size cause changes in the characteristics of the sampling distribution shown below? If so, specifically how does the sampling distribution change?a)The mean of the sampling distribution of
b)The standard deviation of the sampling distribution of c)The shape of the sampling distribution of Unlock Deck
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31
One method for estimating abundance of animals is known as line-intercept sampling. The theory of this method, when applied to Alaskan wolverines, predicts that p = 0.453 of attempts to locate wolverine tracks should be successful. Suppose that biologists will make 100 attempts to locate wolverine tracks in random locations in Alaska.
a)Show that this sample size is large enough to justify using the normal approximation to the sampling distribution of .
b)What is the mean of the sampling distribution of ?
c)What is the standard deviation of the sampling distribution of ?
d)What is the probability that a sample proportion, , would differ from p = 0.453 by as much as 0.05?
a)Show that this sample size is large enough to justify using the normal approximation to the sampling distribution of
.b)What is the mean of the sampling distribution of
?c)What is the standard deviation of the sampling distribution of
?d)What is the probability that a sample proportion,
, would differ from p = 0.453 by as much as 0.05? Unlock Deck
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32
Some biologists believe the evolution of handedness is linked to complex behaviors such as tool-use. Under this theory, handedness would be genetically passed on from parents to children. That is, left-handed parents would be more likely to have left-handed children than right-handed parents. An alternate theory asserts that handedness should be random, with left- and right-handedness equally likely. In a recent study using a simple random sample of n = 76 right-handed parents, 50 of the children born were right-handed. ( = 0.658.) Suppose handedness is a random occurrence with either hand equally likely to be dominant, implying that the probability of a right-handed offspring is p = 0.50.
a)Show that it is reasonable to approximate the sampling distribution of p using a normal distribution.
b)Assuming left- and right-handed children are equally likely from right-handed parents, what is the probability of observing a sample proportion of at least = 0.658?
= 0.658.) Suppose handedness is a random occurrence with either hand equally likely to be dominant, implying that the probability of a right-handed offspring is p = 0.50.a)Show that it is reasonable to approximate the sampling distribution of p using a normal distribution.
b)Assuming left- and right-handed children are equally likely from right-handed parents, what is the probability of observing a sample proportion of at least
= 0.658? Unlock Deck
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33
Suppose that the actual proportion of people at a particular company who use their bicycles to travel to job is 0.35. In a study of bicycle parking places needs, administration would like to estimate this proportion. They plan to take a random sample of 50 employees and use the sample proportion who use bicycles, , as an estimate of the population proportion. Estimate the standard deviation of the sample distribution. Suppose that another sample of different size has been selected, and the standard deviation of this new sample is . Would you expect more or less sample-to-sample variability in the sample proportions than for when ? Is the sample size that resulted in larger or smaller than 50?
A)The standard deviation for the initial sample of the size is .Since the standard deviation is greater than , sample-to-sample variability is less and the new sample size is smaller.
B)The standard deviation for the initial sample of the size is .Since the standard deviation is less than , sample-to-sample variability is more and the new sample size is larger.
C)The standard deviation for the initial sample of the size is .Since the standard deviation is greater than , sample-to-sample variability is more and the new sample size is smaller.
D)The standard deviation for the initial sample of the size is .Since the standard deviation is greater than , sample-to-sample variability is more and the new sample size is smaller.
E)The standard deviation for the initial sample of the size is .Since the standard deviation is greater than , sample-to-sample variability is less and the new sample size is larger.
, as an estimate of the population proportion. Estimate the standard deviation of the sample distribution. Suppose that another sample of different size has been selected, and the standard deviation of this new sample is . Would you expect more or less sample-to-sample variability in the sample proportions than for when ? Is the sample size that resulted in larger or smaller than 50?A)The standard deviation for the initial sample of the size
is .Since the standard deviation is greater than , sample-to-sample variability is less and the new sample size is smaller.B)The standard deviation for the initial sample of the size
is .Since the standard deviation is less than , sample-to-sample variability is more and the new sample size is larger.C)The standard deviation for the initial sample of the size
is .Since the standard deviation is greater than , sample-to-sample variability is more and the new sample size is smaller.D)The standard deviation for the initial sample of the size
is .Since the standard deviation is greater than , sample-to-sample variability is more and the new sample size is smaller.E)The standard deviation for the initial sample of the size
is .Since the standard deviation is greater than , sample-to-sample variability is less and the new sample size is larger. Unlock Deck
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34
How are the quantities, p and , related?
, related? Unlock Deck
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35
The principal at Thomas Jefferson High School has been asked to estimate the proportion of students at KHS who drive to school and use the school parking lot. He takes a random sample of size n = 32 students and calculates a sample proportion, = 0.8. "Now," he exclaims, "since my sample size is greater than 30, the sampling distribution of the sample proportion is approximately normal." Write a short paragraph that explains why his statement is not correct.
= 0.8. "Now," he exclaims, "since my sample size is greater than 30, the sampling distribution of the sample proportion is approximately normal." Write a short paragraph that explains why his statement is not correct. Unlock Deck
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