StatTutor Lesson - The t Distributions

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The t distributions
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      Question 1

      39

      Question 1.

      In practice, what do we use when σ is unknown?

      A.
      B.

      Correct. When we don’t know the value of σ, we estimate it with s, the sample standard deviation.
      Incorrect. When we don’t know the value of σ, we estimate it with s, the sample standard deviation.
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      Question 2

      59

      Question 2.

      Does ¯xμosn¯¯¯xμosn have a standard Normal distribution?

      A.
      B.

      Correct. Because we use “s” to estimate σ, ¯xμosn¯¯¯xμosn does not have a standard Normal distribution. Rather it has a t distribution.
      Incorrect. Because we use “s” to estimate σ, ¯xμosn¯¯¯xμosn does not have a standard Normal distribution. Rather it has a t distribution.
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      Question 3

      123

      Question 3.

      True or false: Every t distribution is specified by its df (degrees of freedom).

      A.
      B.

      Correct. Every t distribution is specified by its degrees of freedom. For t = ¯xμosn¯¯¯xμosn, df = n – 1.
      Incorrect. Every t distribution is specified by its degrees of freedom. For t = ¯xμosn¯¯¯xμosn, df = n – 1.
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      Question 4

      158

      Question 4.

      True or false: A t distribution with 8 degrees of freedom has a smaller mean than the standard Normal distribution.

      A.
      B.

      Correct. Both have the same mean = zero.
      Incorrect. Both have the same mean = zero.
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      Question 5

      179

      Question 5.

      True or false: A t distribution with 3 degrees of freedom has a smaller spread than the standard Normal distribution.

      A.
      B.

      Correct. A t distribution with 3 degrees of freedom has a larger spread than the standard Normal distribution.
      Incorrect. A t distribution with 3 degrees of freedom has a larger spread than the standard Normal distribution.
      2
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      Question 6

      216

      Question 6.

      True or false: As degrees of freedom increase, t density curves get closer to the standard Normal curve.

      A.
      B.

      Correct. The larger the degrees of freedom for a t distribution, the close its density curve is to the standard Normal curve.
      Incorrect. The larger the degrees of freedom for a t distribution, the close its density curve is to the standard Normal curve.
      2
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      Question 7

      355

      Question 7.

      Fill in the blank: A t distribution with 5 degrees of freedom is _____________ a t distribution with 14 degrees of freedom.

      A.
      B.
      C.

      Correct. A t distribution with 5 degrees of freedom has a larger spread than a t distribution with 14 degrees of freedom.
      Incorrect. A t distribution with 5 degrees of freedom has a larger spread than a t distribution with 14 degrees of freedom.
      2
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      Question 8

      390

      Question 8.

      Why is a t distribution with 5 degrees of freedom more spread out than the standard Normal distribution?

      A.
      B.
      C.

      Correct. Because s is an estimate σ, it has error or variability. As a consequence, the t used in inference has to be more spread out than the standard Normal used when we know the value of σ.
      Incorrect. Because s is an estimate σ, it has error or variability. As a consequence, the t used in inference has to be more spread out than the standard Normal used when we know the value of σ.
      2
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      Question 9

      445

      Question 9.

      True or false: We must always use a t distribution whenever we use s to estimate σ.

      A.
      B.

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