8.2    Confidence Intervals When s Unknown: the T-Distribution

Goal:  find confidence interval for m when s is unknown (the usual case).

Approach:  Proceed as before, using S (the sample variance) in place of s.

• previously, used fact that  was standard normal to find confidence interval
• now, look at  to derive interval

Def:    Let , S be the sample mean & variance from sample of size n from a normal population.  Then the random variable

• have a family of distributions, one for each degree of freedom
• graphs of densities:

• symmetric & have general shape of standard normal density, but are broader
• as , density of T_n approaches that of Z
• use info about distribution to find a confidence intervals

95% confidence interval
 

• look at
this quantity has the T-distribution with n-1 degrees of freedom
 
• find interval such that the middle 95% of T-values lie in range:
• let  t_.025  be the value such that    P( T >= t_.025)  =  .025,   i.e., the area under the density curve for T to the right of  t_.025  is .025;  t_.025 is called the upper .025 critical value


• Then
P(-t_.025 <=  T  <= t_.025)  =  .95    (i.e., the probability that T will lie between +t_.025 and -t_.025 is .95)
so

 
i.e., for 95% of samples the value of  will lie in the range
 
• Solving the inequality for  m, we get that for 95% of samples, m will lie in the range

 

• The critical value can be found from (accurate) tables; note that its value depends on  n,  the number of elements in the sample
• Note:  the number of degrees of freedom for a sample of size  n  is  n-1!!

ex:

Sample heights of 40 students; find  = 67.3", S = 3.6".
Then  95% confidence interval is
from tables (p. 732 in text), the .025 critical value for the T distribution with 39 degrees of freedom is
t_.025 = 2.023
gives
or our 95% confidence interval is

The same approach can be used to find confidence intervals for other confidence levels; just use the appropriate critical values. Other commonly used ones are 90% and 99% confidence intervals.
 
 

ex:

90% & 99% confidence intervals for sample of student heights above

90% confidence interval: use the .05 critical value t_.05

from tables, the .05 critical value for the T distribution with 39 degrees of freedom is
t_.05 = 1.685
gives
or

 
99% confidence interval: use the .005 critical value t_.005
from tables, the .005 critical value for the T distribution with 39 degrees of freedom is
t_.005 = 2.708
gives
or

Note:  when n is large, the density function for the T distribution with n-1 degrees of freedom approaches that of the standard normal distribution; thus for large n (n > 100), we can use the critical values from the Z distribution as a good approximation to the values for the T distribution. (In fact, most tables will only give critical values for the T distribution for n up to about 100.)
 
 

ex:

Take a new sample of student heights; sample 200 students, find  = 67.7", S = 4.1"; find a 95% confidence interval for m using this new data.
Interval is
table of critical values in text (p. 732) gives critical values only for n up to 100; the next listed value is for n = *, which gives the critical values for the Z distribution; these can be used for values of n larger than 100. Thus use t_.025 = 1.960, giving
or