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StatTutor Lesson - Sampling Distributions

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Sampling Distributions
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      Question 1

      111

      Question 1.

      True or false: Because the value of ¯x seldom equals µ, and because the value of ¯x varies from one sample to the next, we need to study the sampling distribution of ¯x .

      A.
      B.

      Correct. This is a correct statement.
      Incorrect. This is a correct statement.
      2
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      Questions 2-4

      158

      Question 2.

      How many samples do we need to take in order to estimate the sampling distribution of ¯x?

      A.
      B.
      C.
      D.

      Correct. We generally need a minimum of 2000 samples.
      Incorrect. We generally need a minimum of 2000 samples.
      2
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      Questions 5-6

      239

      Question 5.

      Typically, how many ¯x’s are contained in the sampling distribution of ¯x?

      A.
      B.
      C.

      Correct. Even though it is called the “sampling distribution of ¯x," it is the distribution of all ¯x’s from all possible samples.
      Incorrect. Even though it is called the “sampling distribution of ¯x," it is the distribution of all ¯x’s from all possible samples.
      2
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      Question 7

      329

      Question 7.

      Why do we approximate the sampling distribution of ¯x rather than create the actual sampling distribution of ¯x?

      A.
      B.
      C.

      Correct. We actually need fewer samples and we don’t have to list all possible samples. Taking two thousand samples is usually enough to get a good estimation of the actual sampling distribution of ¯x.
      Incorrect. We actually need fewer samples and we don’t have to list all possible samples. Taking two thousand samples is usually enough to get a good estimation of the actual sampling distribution of ¯x.
      2
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      Question 8

      417

      Question 8.

      If we were to take a sample to estimate the parameter of interest, in context what is that parameter?

      A.
      B.
      C.

      Correct. The mean of all the individuals in the population is the parameter. In this case, all individuals are all stocks on the New York stock exchange. The measurement on each stock is the closing price. So the parameter in context is the mean closing price of all New York Stock Exchange stocks.
      Incorrect. The mean of all the individuals in the population is the parameter. In this case, all individuals are all stocks on the New York stock exchange. The measurement on each stock is the closing price. So the parameter in context is the mean closing price of all New York Stock Exchange stocks.
      2
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      Questions 9-10

      469

      Question 9.

      How many samples are we going to take?

      A.
      B.
      C.

      Correct. Don’t confuse the sample size with the number of samples we are going to take. We need to take many, many samples of size 4 and find the mean of the four prices for each sample of size four.
      Incorrect. Don’t confuse the sample size with the number of samples we are going to take. We need to take many, many samples of size 4 and find the mean of the four prices for each sample of size four.
      2
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      Question 11

      488

      Question 11.

      Does this sample mean ¯x = $28.22 constitute a sampling distribution of ¯x?

      A.
      B.

      Correct. One sample mean is not a sampling distribution of ¯x; we need all possible ¯x’s for the sampling distribution of ¯x.
      Incorrect. One sample mean is not a sampling distribution of ¯x; we need all possible ¯x’s for the sampling distribution of ¯x.
      2
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      Question 12

      526

      Question 12.

      Is the list of ¯x’s on the bottom right of the slide an estimated sampling distribution of ¯x?

      A.
      B.

      Correct. ¯x-values from many many samples estimate the sampling distribution of ¯x.
      Incorrect. ¯x-values from many many samples estimate the sampling distribution of ¯x.
      2
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      Questions 13-14

      546

      Question 13.

      In constructing the histogram of ¯x’s, we began with circles. What did each circle represent?

      A.
      B.
      C.

      Correct. Since we are constructing a histogram of ¯x’s, each circle represents the value of one ¯x.
      Incorrect. Since we are constructing a histogram of ¯x’s, each circle represents the value of one ¯x.
      2

      Question 15

      563

      Question 15.

      Fill in the blank: The center of the estimated sampling distribution of ¯x in red on the right is ___________ the mean of population of closing stock prices displayed in blue on the left.

      A.
      B.
      C.

      Correct. The mean of the sampling distribution of ¯x we need all possible ¯x equals the mean of the population.
      Incorrect. The mean of the sampling distribution of ¯x we need all possible ¯x equals the mean of the population.
      2
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      Question 16

      583

      Question 16.

      Fill in the blank: The spread of the estimated sampling distribution of ¯x in red on the right is ___________ the spread of population of closing stock prices displayed in blue on the left.

      A.
      B.
      C.

      Correct. Sample means are typically closer to μ than measurements on individuals. Thus, sample means (displayed in the red histogram on the right) are less variable than individual measurements (displayed in the blue histogram on the left).
      Incorrect. Sample means are typically closer to μ than measurements on individuals. Thus, sample means (displayed in the red histogram on the right) are less variable than individual measurements (displayed in the blue histogram on the left).
      2
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      Question 17

      590

      Question 17.

      Fill in the blank: The shape of the estimated sampling distribution of ¯x in red on the right is ___________ the shape of population of closing stock prices displayed in blue on the left?

      A.
      B.
      C.

      Correct. Averaging the sample means tends to average out the skewness. The bigger the sample size, the more skewness is averaged out and the more symmetrically the sample means cluster about μ.
      Incorrect. Averaging the sample means tends to average out the skewness. The bigger the sample size, the more skewness is averaged out and the more symmetrically the sample means cluster about μ.
      2
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      Questions 18-20

      635

      Question 18.

      Do you think the center of the sampling distribution of ¯x created from samples of size 16 will be different from $26?

      A.
      B.

      Correct.
      Incorrect.
      2
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      Question 21

      719

      Question 21.

      Fill in the blank: The center of the estimated sampling distribution of ¯x in red on the right is ___________ the mean of population of closing stock prices displayed in blue on the left.

      A.
      B.
      C.

      Correct. The mean of the sampling distribution of ¯x equals the mean of the population.
      Incorrect. The mean of the sampling distribution of ¯x equals the mean of the population.
      2
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      Question 22

      750

      Question 22.

      Fill in the blank: The spread of the estimated sampling distribution of ¯x in red on the right is ___________ the spread of population of closing stock prices displayed in blue on the left.

      A.
      B.
      C.

      Correct. Sample means are typically closer to μ than measurements on individuals. Thus, as with samples of size 4, sample means (displayed in the red histogram on the right) are less variable than individual measurements (displayed in the blue histogram on the left).
      Incorrect. Sample means are typically closer to μ than measurements on individuals. Thus, as with samples of size 4, sample means (displayed in the red histogram on the right) are less variable than individual measurements (displayed in the blue histogram on the left).
      2
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      Question 23

      764

      Question 23.

      Fill in the blank: The shape of the estimated sampling distribution of ¯x in red on the right is ___________ the shape of population of closing stock prices displayed in blue on the left?

      A.
      B.
      C.

      Correct. The averaging process used to compute sample means tends to average out skewness. The bigger the sample size, the more skewness is averaged out and the more symmetrically the sample means cluster about μ. Thus, the histogram of sample means from samples of size 16 is closer to Normal in shape than the histogram of closing stock prices.
      Incorrect. The averaging process used to compute sample means tends to average out skewness. The bigger the sample size, the more skewness is averaged out and the more symmetrically the sample means cluster about μ. Thus, the histogram of sample means from samples of size 16 is closer to Normal in shape than the histogram of closing stock prices.
      2
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      Question 24

      813

      Question 24.

      Fill in the blank: The mean of the sampling distribution of ¯x created from samples of size 4 is ______________ the mean of the sampling distribution of ¯x created from samples of size 16.

      A.
      B.
      C.

      Correct. Since the mean of the sampling distribution of ¯x equals the mean of the population, the means of all sampling distributions are equal.
      Incorrect. Since the mean of the sampling distribution of ¯x equals the mean of the population, the means of all sampling distributions are equal.
      2
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      Question 25

      869

      Question 25.

      Fill in the blank: The spread of the sampling distribution of ¯x created from samples of size 4 is ______________ the spread of the sampling distribution of ¯x created from samples of size 16.

      A.
      B.
      C.

      Correct. Since the larger the sample size, the closer the sample means are to µ, sample means from larger samples have less spread than sample means from smaller samples.
      Incorrect. Since the larger the sample size, the closer the sample means are to µ, sample means from larger samples have less spread than sample means from smaller samples.
      2
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      Question 26

      889

      Question 26.

      Fill in the blank: The shape of the sampling distribution of ¯x created from samples of size 4 is ______________ the shape of the sampling distribution of ¯x created from samples of size 16.

      A.
      B.
      C.

      Correct. The averaging process used to compute sample means tends to average out skewness. The bigger the sample size, the more skewness is averaged out and the more symmetrically the sample means cluster about μ. The histogram of sample means from samples of size 16 is closer to Normal in shape than the histogram of sample means from samples of size 4.
      Incorrect. The averaging process used to compute sample means tends to average out skewness. The bigger the sample size, the more skewness is averaged out and the more symmetrically the sample means cluster about μ. The histogram of sample means from samples of size 16 is closer to Normal in shape than the histogram of sample means from samples of size 4.
      2
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      Question 27

      925

      Question 27.

      What happens to the means of the four sampling distributions of ¯x created from samples of size 4, 16, 32 and 64 as sample size increases?

      A.
      B.
      C.

      Correct. The mean of the sampling distribution of ¯x equals the mean of the population which is 26.
      Incorrect. The mean of the sampling distribution of ¯x equals the mean of the population which is 26.
      2
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      Question 28

      946

      Question 28.

      What happens to the spreads of the four sampling distributions of ¯x created from samples of size 4, 16, 32 and 64 as sample size increases?

      A.
      B.
      C.

      Correct. Since the larger the sample size, the closer the sample means are to µ, sample means from larger samples have less spread than sample means from smaller samples. So, as sample size increases the spread of the sampling distribution of ¯x decreases.
      Incorrect. Since the larger the sample size, the closer the sample means are to µ, sample means from larger samples have less spread than sample means from smaller samples. So, as sample size increases the spread of the sampling distribution of ¯x decreases.
      2
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      Question 29

      995

      Question 29.

      What happens to the shape of the four sampling distributions of ¯x created from samples of size 4, 16, 32 and 64 as sample size increases?

      A.
      B.
      C.

      Correct. The averaging process used to compute sample means tends to average out skewness. The bigger the sample size, the more skewness is averaged out and the more symmetrically the sample means cluster about μ. So, as sample size increases, the shape of the sampling distribution of ¯x gets more Normal.
      Incorrect. The averaging process used to compute sample means tends to average out skewness. The bigger the sample size, the more skewness is averaged out and the more symmetrically the sample means cluster about μ. So, as sample size increases, the shape of the sampling distribution of ¯x gets more Normal.
      2
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      Question 30

      1076

      Question 30.

      How will we compute probabilities on ¯x?

      A.
      B.

      Correct. Since the shape of the sampling distribution of ¯x is approximately Normal for large sample sizes, we will take a large sample and use a Normal curve.
      Incorrect. Since the shape of the sampling distribution of ¯x is approximately Normal for large sample sizes, we will take a large sample and use a Normal curve.
      2
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