The distribution of T is not a standard normal dis- tribution, although for large n, and even moder- ately sized n, it’s very close to a standard normal distribution. The distribution of T is called Stu- dent’s t-distribution. On page 180 of the text, the standard normal distribution is drawn along with The t and F distributions Student’s t-distribution for n = 2 and n = 10. Math 218, Mathematical Statistics Since for large values of n, the t-distribution is so D Joyce, Spring 2016 close to the standard normal distribution, the T - distribution is only needed for n small, say n ≤ 30. Student’s t-distribution and Snedecor-Fisher’s F - It turns out that the ratio between T and Z (the distribution. These are two distributions used in scaled sample mean described above) is the square statistical tests. The first one is commonly used to root of a scaled χ2 distribution. Precisely, estimate the mean µ of a normal distribution when the variance σ2 is not known, a common situation. Z X − µX − µ S = √ √ = , The second is a rather special purpose distribution T σ/ n S/ n σ used to estimate the ratio of the variances of two 2 2 normal distributions. σ χn−1 and, as we saw last time, S2 ∼ , so S/σ n − 1 S2 χ2 Student’s t-distribution. We know that if a is the square root of ∼ n−1 . That’s enough sample is drawn from a normal distribution with σ2 n − 1 information to compute a t-distribution. Table A.4 mean µ and variance σ2 that the scaled sample av- has critical values for the t-distribution. erage We’ll work through example 5.6 in class. X − µ Z = √ σ/ n Snedecor-Fisher’s F -distribution. This dis- is a standard normal distribution. That informa- tribution is used to estimate the ratio of the vari- tion can be used to determine just how good X ances of two normal distributions. If we have is as an estimator for the unknown value µ. Un- two random samples, the first X1,...,Xn1 from a fortunately, σ2 is also an unknown. (At least n is N(µ1, σ1) distribution, while the second Y1,...,Yn2 known.) So, what can we do? from a N(µ2, σ2) distribution, then we know their Student (the nom de plume of William Sealey sample variances are scaled χ2 distributions. That Gosset (1876–1937)) determined that the unknown is, σ could be replaced by the sample standard de- Pn1 2 σ2χ2 2 i=1(Xi − X) 1 n1−1 viation S, which is the square root of the sample S1 = ∼ n1 − 1 n1 − 1 variance and n n P 2 (Y − X)2 σ2χ2 1 X 2 i=1 i 2 n2−1 S2 = (X − X)2. S2 = ∼ . n − 1 i n2 − 1 n2 − 1 i=1 Therefore, for the ratio of these sample variances That results in a statistic that looks something like we have S2 σ2χ2 σ2χ2 the scaled sample average, but isn’t identical. It’s 1 1 n1−1 2 n2−1 2 ∼ , the variable T defined by S2 n1 − 1 n2 − 1 and so 2 2 2 X − µ S1 /σ1 χn1−1/(n1 − 1) T = √ . 2 2 ∼ 2 . S/ n S2 /σ2 χn2−1/(n2 − 1)

1 The random variable on the right, a scaled ratio of two χ2 distributions, has what is called a Snedecor- Fisher F -distribution with n1 − 1 degrees of free- dom in the numerator and n2 − 1 degrees of free- dom in the denominator. The notation for an F - distribution with ν1 and ν2 degrees of freedom is

Fν1,ν2 . Table A.6 has critical values for this F dis- tribution. Statistical tests based on the F -distribution can determine if the variances σ1 and σ2 of two normal distributions are the same by looking at the ratio 2 2 of two sample variances S1 /S2 .

Math 218 Home Page at http://math.clarku.edu/~djoyce/ma218/

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