StatTutor Lesson - The Central Limit Theorem

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The Central Limit Theorem
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      Question 1

      46

      Question 1.

      What is the definition of the sampling distribution of x¯?

      A.
      B.
      C.

      Correct. A distribution of a variable gives the possible values of the variable together with how often each value occurs. So, the sampling distribution of x¯ gives the possible values of x¯ together with how often each value occurs.
      Incorrect. A distribution of a variable gives the possible values of the variable together with how often each value occurs. So, the sampling distribution of x¯ gives the possible values of x¯ together with how often each value occurs.
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      Question 2

      65

      Question 2.

      If a large population has mean, µ = 40 and standard deviation, σ = 12, what is the mean of the sampling distribution of x¯ created from all possible samples of size n = 9?

      A.
      B.
      C.
      D.
      E.
      F.

      Correct. The mean of the sampling distribution of x¯ always exactly equals μ, so the mean equals µ = 40.
      Incorrect. The mean of the sampling distribution of x¯ always exactly equals μ, so the mean equals µ = 40.
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      Question 3

      77

      Question 3.

      If a large population has mean, µ = 40 and standard deviation, σ = 12, what is the standard deviation of the sampling distribution of x¯ created from all possible samples of size n = 9?

      A.
      B.
      C.
      D.
      E.
      F.

      Correct. The standard deviation of the sampling distribution of equals σn regardless of sample size and shape of the population, so σn = 129 = 4.0.
      Incorrect. The standard deviation of the sampling distribution of equals σn regardless of sample size and shape of the population, so σn = 129 = 4.0.
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      Question 4

      111

      Question 4.

      If a large population has mean, µ = 40 and standard deviation, σ = 12, what is the shape of the sampling distribution of x¯ created from all possible samples of size n = 9?

      A.
      B.
      C.

      Correct. For the sampling distribution to be Normal, the population shape must be Normal; for the sampling distribution of x¯ to be approximately Normal, the sample size must be large. Neither of those conditions are met in the description of the problem.
      Incorrect. For the sampling distribution to be Normal, the population shape must be Normal; for the sampling distribution of x¯ to be approximately Normal, the sample size must be large. Neither of those conditions are met in the description of the problem.
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      Questions 5-8

      282

      Question 6.

      According to the Central Limit Theorem, what has an approximate Normal shape if the sample is large and random?

      A.
      B.
      C.

      Correct. The Central Limit Theorem says that the shape of the sampling distribution of x¯ is approximately Normal if a large random sample is taken. So the Central Limit theorem has to do with the shape of the sampling distribution of x¯, not the sample and not the population.
      Incorrect. The Central Limit Theorem says that the shape of the sampling distribution of x¯ is approximately Normal if a large random sample is taken. So the Central Limit theorem has to do with the shape of the sampling distribution of x¯, not the sample and not the population.
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      Questions 9-10

      302

      Question 10.

      True or false: The mean of the sampling distribution of x¯ equals μ only when n > 30.

      A.
      B.

      Correct. The mean of the sampling distribution of x¯ equals µ for all sample sizes.
      Incorrect. The mean of the sampling distribution of x¯ equals µ for all sample sizes.
      2
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      Questions 11-14

      463

      Question 12.

      True or false: The mean of the sampling distribution of x¯ equals μ regardless of sample size.

      A.
      B.

      Correct. This is a true statement.
      Incorrect. This is a true statement.
      2
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      Question 15

      530

      Question 16.

      When sampling from a non-Normal population with a large random sample, what z-score formula should we use to find a probability on x¯?

      A.
      B.
      C.

      Correct. When finding a probability on x¯, we must use the standard deviation of the sampling distribution of x¯ (namely σn) in the denominator. Thus, the appropriate z-score is z=x¯μσn.
      Incorrect. When finding a probability on x¯, we must use the standard deviation of the sampling distribution of x¯ (namely σn) in the denominator. Thus, the appropriate z-score is z=x¯μσn.
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      Questions 16-17

      612

      Question 17.

      Which z-score formula should we use to find a probability on an individual bottle?

      A.
      B.
      C.

      Correct. When finding a probability on an individual x, we must use the standard deviation of population (namely, σ) in the denominator. Thus, the appropriate z-score is z=xμσ.
      Incorrect. When finding a probability on an individual x, we must use the standard deviation of population (namely, σ) in the denominator. Thus, the appropriate z-score is z=xμσ.
      2
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      Questions 18-19

      698

      Question 18.

      After looking up the z-score in the standard Normal table, why did we subtract the table probability from 1.0?

      A.
      B.

      Correct. The standard Normal table gives area on the left or “less than” probabilities. We needed to find the probability that a randomly selected bottle weighs “more than” 1.1 pounds which is area on the right.
      Incorrect. The standard Normal table gives area on the left or “less than” probabilities. We needed to find the probability that a randomly selected bottle weighs “more than” 1.1 pounds which is area on the right.
      2
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      Question 20

      776

      Question 20.

      Which z-score formula should we use to find a probability on the mean weight of a random sample of eight bottles?

      A.
      B.
      C.

      Correct. When finding a probability on x¯, we must use the standard deviation of the sampling distribution of x¯ (namely σn) in the denominator. Thus, the appropriate z-score is z=x¯μσn.
      Incorrect. When finding a probability on x¯, we must use the standard deviation of the sampling distribution of x¯ (namely σn) in the denominator. Thus, the appropriate z-score is z=x¯μσn.
      2
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      Question 21

      896

      Question 21.

      Why could we use the standard Normal table to find a probability on the mean of a random sample of eight bottles weighing more than 1.1 pounds?

      A.
      B.
      C.

      Correct. Since the population distribution is Normal, the sampling distribution of x¯ is also Normal for all sample sizes. Thus, we could use the standard Normal table to find the probability on the mean weight from a random sample of eight bottles.
      Incorrect. Since the population distribution is Normal, the sampling distribution of x¯ is also Normal for all sample sizes. Thus, we could use the standard Normal table to find the probability on the mean weight from a random sample of eight bottles.
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      Question 22

      929

      Question 22.

      True or false: We could compute both a probability on the weight of a randomly selected individual bottle and a probability on the mean weight of a random sample of eight bottles using the standard Normal table because weights of the bottles is Normally distributed.

      A.
      B.

      Correct. Whenever the population distribution is Normal, we can compute a probability on an individual x and a probability on a sample mean using the standard Normal table.
      Incorrect. Whenever the population distribution is Normal, we can compute a probability on an individual x and a probability on a sample mean using the standard Normal table.
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      Question 23

      970

      Question 23.

      Can we use a Normal distribution to fine a probability on the closing price of an individual stock?

      A.
      B.

      Correct. We can only compute a probability on an individual when the population distribution is Normal. The population distribution of closing stock price is right skewed, not Normal so we cannot use a Normal distribution to find a probability on the closing price of an individual stock.
      Incorrect. We can only compute a probability on an individual when the population distribution is Normal. The population distribution of closing stock price is right skewed, not Normal so we cannot use a Normal distribution to find a probability on the closing price of an individual stock.
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      Question 24

      1062

      Question 24.

      Why is the sampling distribution of x¯ approximately Normal?

      A.
      B.
      C.

      Correct. Since the sample size is 32 and the sample is random, we can apply the Central Limit Theorem and say that the sampling distribution of x¯ is approximately Normal.
      Incorrect. Since the sample size is 32 and the sample is random, we can apply the Central Limit Theorem and say that the sampling distribution of x¯ is approximately Normal.
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      Question 25

      1115

      Question 25.

      Which z-score formula should we use to find a probability on the mean closing price of a random sample of thirty-two stocks?

      A.
      B.
      C.

      Correct. When finding a probability on x¯ (sample mean), we must use the standard deviation of the sampling distribution of x¯ (namely σn) in the denominator. Thus, the appropriate z-score is z=x¯μσn.
      Incorrect. When finding a probability on x¯ (sample mean), we must use the standard deviation of the sampling distribution of x¯ (namely σn) in the denominator. Thus, the appropriate z-score is z=x¯μσn.
      2
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      Questions 26-27

      1175

      Question 26.

      Why is the probability found by looking up the z-score of –3.11 in the standard Normal table the answer?

      A.
      B.

      Correct. The standard Normal table gives area on the left or “less than” probabilities. Since we want a “less than” probability, the number found by looking up z = –3.11 is the correct answer.
      Incorrect. The standard Normal table gives area on the left or “less than” probabilities. Since we want a “less than” probability, the number found by looking up z = –3.11 is the correct answer.
      2
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      Question 28

      1264

      Question 28.

      True or false: Whenever the sampling distribution of x¯ is either Normal or approximately Normal and the sample is random, we can use z=x¯μσn and a Normal distribution of find a probability on a sample mean.

      A.
      B.

      Correct. This is a true statement and will be used repeatedly when we do inference.
      Incorrect. This is a true statement and will be used repeatedly when we do inference.
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      Question 29

      1337

      Question 29.

      Why don’t we need to create the sampling distribution of x¯ before computing a probability on x¯?

      A.
      B.
      C.

      Correct. While the shape of a sampling distribution of x¯ is not always Normal, it is when either the population shape is Normal or the sample size is large. Further, the mean of the sampling distribution of x¯ always equals µ and the standard deviation of the sampling distribution of x¯ equals σn. These facts allow us to compute a probability on x¯ without creating the sampling distribution of x¯.
      Incorrect. While the shape of a sampling distribution of x¯ is not always Normal, it is when either the population shape is Normal or the sample size is large. Further, the mean of the sampling distribution of x¯ always equals µ and the standard deviation of the sampling distribution of x¯ equals σn. These facts allow us to compute a probability on x¯ without creating the sampling distribution of x¯.
      2
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      Questions 30-32

      1484

      Question 30.

      Which graph displays the distribution of all closing stock prices?

      A.
      B.
      C.

      Correct. The population consists of all stocks so the histogram of all closing stock prices is a histogram of the population.
      Incorrect. The population consists of all stocks so the histogram of all closing stock prices is a histogram of the population.
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      Question 33-35

      1562

      Question 33.

      What symbol represents the mean of a population?

      A.
      B.
      C.
      D.
      E.

      Correct. μ is the symbol for the mean of a population.
      Incorrect. μ is the symbol for the mean of a population.
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      Questions 36-38

      1577

      Question 36.

      What symbol represents the standard deviation of a population?

      A.
      B.
      C.
      D.
      E.

      Correct. σ represents the standard deviation of a population.
      Incorrect. σ represents the standard deviation of a population.
      2
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      Questions 39-42

      1634

      Question 39.

      If the shape of the population distribution is Normal, what is the shape of the histogram of data in a sample?

      A.
      B.
      C.

      Correct. If the sample size is large, the shape of the histogram of data in a sample will be approximately Normal and gets more Normal as sample size increases. If sample size is small, we cannot predict the shape.
      Incorrect. If the sample size is large, the shape of the histogram of data in a sample will be approximately Normal and gets more Normal as sample size increases. If sample size is small, we cannot predict the shape.
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      Questions 43-44

      1686

      Question 43.

      When can we compute a probability on an individual x using a Normal distribution?

      A.
      B.
      C.

      Correct. Since we want to compute a probability on an individual x, we can only do this if the population distribution is Normal.
      Incorrect. Since we want to compute a probability on an individual x, we can only do this if the population distribution is Normal.
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