Lecture 8 - The Students t Distribution What do we do if...

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The Student’s t Distribution What do we do if (a) we don’t know σ and (b) n is small? If the population of interest is normally distributed, we can use the Student’s t-distribution in place of the standard normal distribution. William S. Gossett who developed the t-distribution, wrote under the name “Student” since, as an employee of the Guiness Brewery in Dublin, he was required by the firm to use a pseudonym in publishing his results. We will use Z for EITHER (1) known σ OR (2) large n Use t for (1) Small sample AND (2) Taken from N.D. population* AND (3) Unkown σ * What can we do if the population is not normally distributed and we have a small sample? Always use non-parametric statistical methods in this case. The t-distribution looks like the normal distribution, except that it is more spread out. It is still symmetrical about the mean; mean=median=mode; goes from -∞ to +∞. Student’s t distribution is not a single distribution as is the standardized normal distribution (Z), but rather it is a series of distributions, one for each number of degrees of freedom. As n gets larger, student’s t distribution approaches the normal distribution. t statistic for testing hypotheses (n-1 degrees of freedom): Degrees of Freedom:
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# of degrees of freedom = the sample size minus 1. We “lose” a degree of freedom each time a statistic computed from the sample is used as a point estimator of a parameter. In this case, we do not know the population σ, so instead of: we use s to estimate σ : [s is an unbiased estimator of σ .] We use - a statistic - instead of μ in the formula for s. The price we pay is a loss of a degree of freedom. Notice that this penalty increases the size of the standard deviation. Simple example: If =3 and when we have 5 numbers. 1, 2, 3, 4, __. The 5 th number must be 5 since =3 (which means the total must be 15). You only have leeway with 4 numbers—loss of 1 d.f.
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The Short t Table: Critical Values of t For a particular number of degrees of freedom, each entry represents the critical value of t
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Lecture 8 - The Students t Distribution What do we do if...

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