11tdistribution

# 11tdistribution - Statistical Techniques I EXST7005 The t...

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Statistical Techniques I EXST7005 Still here The t distribution

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The t-test of Hypotheses the t distribution is used the same as Z distribution, except it is used where sigma ( σ ) ,i unknown (or where f8e5 Y is used instead of μ to calculate deviations) ti = ( Yi - f8e5 Y )/ S t = ( f8e5 Y - μ 0)/ S f8e5 Y = ( f8e5 Y - μ 0)/( S/ n ) where; S = the sample standard deviation, (calculated using f8e5 Y) S f8e5 Y = the sample standard error
The t-test of Hypotheses (continued) E(t) = 0 The variance of the t distribution is greater than that of the Z distribution (except where n ), since S2 estimates σ 2, but is never as good (reliability is less) Z distribution t distribution mean 0 0 variance 1 1

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CHARACTERISTICS OF THE t DISTRIBUTION It is symmetrically distributed about a mean of 0 t ranges to ± (i.e. - t + ) There is a different t distribution for each degre of freedom (df), since the distribution changes a the degrees of freedom change. It has a broader spread for smaller df, and narrow (approaching the Z distribution) as df increase
CHARACTERISTICS OF THE t DISTRIBUTION (continued) 4) As the df ( γ , gamma) approaches infinity ( ) the t distribution approaches the Z distribution. For example; Z (no df associated); middle 95% is between ± 1.96 t with 1 df; middle 95% is between ± 12.706 t with 10 df; middle 95% is between ± 2.228 t with 30 df; middle 95% is between ± 2.042 t with df; middle 95% is between ± 1.96

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Tables in general The tables we will use will ALL be giving the are in the tail ( α ). However, if you examine a number of tables you will find that this is not always true. Even when it is true, some tables will give the value of α as if it were in two tails, and some as if it were in one tail. For example, we want to conduct a two- tailed Z test at the α =0.05 level. We happen to know that Z=1.96.
Tables in general (continued) If we look at this value in the Z tables we expec to see a value of 0.025, or α /2. But many table would show the probability for 1.96 as 0.975, an some as 0.05. Why the difference? I just depends on how the tables are presented. Some of the alternatives are shown below.

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Tables in general (continued) Table gives cumulative distribution starting at - infinity. You want to find the probability corresponding to 1- α /2. -4 -3 -2 -1 0 1 2 3 4 1- α= 0.975 α= 0.02 5 Table value, 0.975
(continued) Some tables may also start at zero (0.0) and give the cumulative area from this point. This would be less common. The value that leaves 025 in the upper tail would be 0.475. Among the tables like ours, that give the area in

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11tdistribution - Statistical Techniques I EXST7005 The t...

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