1. What if the population standard deviation, , is unknown? We could estimate it by the sample standard deviation s which is the square root of the sample variance But what becomes of the standardized mean when s estimates  ? It now becomes a t-distribution with n-1 degrees of freedom. Some assumptions: –the population from which the sample is taken must be normal –the sample is a random sample (so the X’s are i.i.d.)

2. Properties of t-distribution w/ degrees of freedom (df): –symmetric around zero (mean is zero) –“bell shaped” like the normal but with a little more area in the tails –standard deviation depends upon the degrees of freedom: s.d.=Note: this exceeds 1 but as the sample size increases, s.d. approaches 1. In fact, it can be shown that as n increases to infinity, the t-distribution with  degrees of freedom approaches the standard normal distribution. As a rule of thumb, you may use the normal approximation when n is larger than 30 or so... see Table 4, last row, compared with previous rows... –Tail probabilities can be found in Table 4 in the back of the book (or you may use the TI-83: (tcdf is the function under 2 nd vars...). These are denoted by t  - see Table 4... –the normal population assumption on the previous slide is not too restrictive... can you simulate with R to see this??

3. Now let’s look at the sampling distribution of the sample variance s 2 when the X’s come from a normal population...Theorem 6.4 shows that a simple function of the sample variance has a chi-square distribution with n-1 degrees of freedom. The chi-square density is just a gamma density with alpha=(#df/2) and beta=2. Here  = n-1 = # d.f. Properties of the chi-square distribution: –non-symmetric; P(  chi  square < 0) = 0. –use the gamma  2, 2) distribution to see that the mean of a chi- square with  degrees of freedom is  and the variance is 2 

4. The final distribution in section 6.4 is the F-distribution which arises as the quotient of two chi-squares, but we generally see it as the quotient of two sample variances where the numerator df=n 1 – 1= 1  and denominator df=n 2 – 1= 2 . As with normal, t, and chi-square, the F-distribution is tabulated in Table 6 (and on the TI-83: Fcdf) HW: Finish reading section 6.4 –use R to get plots of the density curves of various t, chi-square and F distributions... I’ll get you started in class today. –do # 6.20-6.26 on page 221. –do # 6.31-6.33, 6.35, 6.36 on pages 223-224.