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30 Cards in this Set
- Front
- Back
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1. Suppose X ~ N(0, 1). Which, if either, is more likely: .3 ≤ X ≤ .4, or .7 ≤ X ≤ .8? Explain your answer (you do not need to calculate these values).
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1
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2. Suppose X ~ N(0, 1). Which, if either, is more likely: .3 < X < .4, or .7 ≤ X ≤ .8? Explain your answer (you do not need to calculate these values).
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2
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3. Suppose X ~ N(0, 1). Which, if either, is more likely: –1 ≤ X ≤ 1, or 5 ≤ X ≤ 8? Explain your answer (you do not need to calculate these values).
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3
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4. Write out the mathematical formula for the pdf for N(μ, σ2).
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4
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5. Draw and label the graphs of the pdf (together, on one set of axes) of N(0,1) and N(-1, 2). Your graphs should display how the different parameters of the two distributions affect the pdfs.
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5
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6. Draw and label the graphs of the cdf (together, on one set of axes) of N(0,1) and N(-1, 2). Your graphs should display how the different parameters of the two distributions affect the cdfs.
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6
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7. Explain what the Central Limit Theorem says. Be sure to include the background conditions necessary for the theorem to hold, and the “limit statement” as presented in the book that gives the mathematical essence of the theorem.
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7
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8. Give the limit formula in the Central Limit Theorem, not as it is presented in the book, but as a limit formula of a standardization of a random variable. In this formulation, what random variable is being standardized?
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8
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9. Show that the two forms of the Central Limit Theorem are equivalent by calculating that:
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9
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10. What is the total area under the curve of the pdf of N(0, 1)?
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10
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11. What is E[X] if X ~ N(μ, σ2)?
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11
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12. What is E[X] if X ~ N(1, 3)?
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12
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13. What is E[(X – E[X])2] if X ~ N(μ, σ2)?
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13
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14. What is E[(X – E[X])2] if X ~ N(1, 3)?
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14
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15. What is α1 if X ~ N(μ, σ2)?
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15
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16. What is α1 if X ~ N(1, 3)?
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16
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17. What is α2 if X ~ N(μ, σ2)?
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17
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18. What is α2 if X ~ N(1, 3)?
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18
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19. Suppose Z is the standardization of X (Z = ), where μ and σ are the mean and standard deviation of X). Calculate the mean of Z.
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19
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20. Suppose Z is the standardization of X (Z = ), where μ and σ are the mean and standard deviation of X). Calculate the standard deviation of Z.
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20
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21. Suppose that Y = a + bX (b > 0). Calculate to show that the standardization of Y is the standardization of X.
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21
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22. Suppose that Y = 4 + 2X, where the mean of X is 8 and the standard deviation of X is 3. Calculate the standardizations of X and Y separately.
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22
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23. What is the standardized skew of N(μ, σ2)?
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23
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24. What is the standardized kurtosis of N(μ, σ2)?
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24
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25. How much of the probability of an N(μ, σ2) distribution is within 1.96 standard deviations of the mean?
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25
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26. If X ~N(μ, σ2), what is what is the probability that X ≤ (μ + 1.64σ)?
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26
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27. Why does the normal distribution occur frequently in nature?
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27
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28. Give three real life examples that are probably (nearly) normally distributed; for each one, write a sentence or two about why they probably are normally distributed.
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28
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29. For N(μ, σ2), calculate the largest value of its pdf. Show also that any other value will be less than this number.
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29
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30. For N(40, 18), calculate the largest value of its pdf. Show also that any other value will be less than this number.
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30
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