Download chi-square distribution with n

• What if the population standard deviation, s, is
unknown? We could estimate it by the sample standard
deviation s which is the square root of the sample
n
variance
( X  X )2
s 
2

i 1
i
n 1
• But what becomes of the standardized mean when s
estimates s?
X 
X 

s
s
n
n
• 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.)
• Properties of t-distribution w/ n 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:
n
s.d.=
Note: this exceeds 1 but as the sample size
n 2
increases, s.d. approaches 1. In fact, it can be shown that as n
increases to infinity, the t-distribution with n 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 2nd vars...).
These are denoted by ta - see Table 4...
– the normal population assumption on the previous slide is not too
restrictive... can you simulate with R to see this??
• Now let’s look at the sampling distribution of the sample
variance s2 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.
n 1
s  n
2
2
Here n = n-1 = # d.f.
• Properties of the chi-square distribution:
s
2
– non-symmetric; P( chisquare < 0) = 0.
– use the gamma(n/2, 2) distribution to see that the mean of a chisquare with n degrees of freedom is n and the variance is 2n.
• 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
Fn1 ,n 2
s12
 2
s2
where the numerator df=n1 – 1= n1 and
denominator df=n2 – 1= n2 .
• 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.