we have
n¼ 31
x¼ 84 : 65
s¼ 24 : 00leading to a standard error
SEðxÞ¼24 : 00
ffiffiffiffiffi
31p¼ 4 : 31
and a 95% confidence interval for the population mean:
84 : 65 Gð 1 : 96 Þð 4 : 31 Þ¼ð 76 : 2 ; 93 : 1 Þ(The resulting interval is wide, due to a large standard deviation as observed
from the sample,s¼ 24 :0, reflecting heterogeneity of sample subjects.)
4.2.2 Uses of Small Samples
The procedure for confidence intervals in Section 4.2.1 is applicable only to
large samples (say,n>25). For smaller samples, the results are still valid if the
population variances^2 is known and standard error is expressed ass=
ffiffiffi
np
.
However,s^2 is almost always unknown. Whensis unknown, we can estimate
it bysbut the procedure has to be modified by changing the coe‰cient to be
multiplied by the standard error to accommodate the error in estimatingsbys;
how much larger the coe‰cient is depends on how much information we have
in estimatings(bys), that is, the sample sizen.
Therefore, instead of taking coe‰cients from the standard normal distribu-
tion table (numbers such as 2.576, 1.960, 1.645, and 1.282 for degrees of confi-
dence 99%, 95%, 90%, and 80%), we will use corresponding numbers from thet
curves where the quantity of information is indexed by the degree of freedom
ðdf¼n 1 Þ. The figures are listed in Appendix C; the column to read is the
one with the correct normal coe‰cient on the bottom row (marked with
df¼y). See, for example, Table 4.5 for the case where the degree of confi-
dence is 0.95.(For better results, it is always a good practice to use the t table
regardless of sample size because coe‰cients such as 1.96 are only for very large
sample sizes.)
Example 4.4 In an attempt to assess the physical condition of joggers, a sam-
ple ofn¼25 joggers was selected and maximum volume oxygen (VO 2 ) uptake
was measured, with the following results:
156 ESTIMATION OF PARAMETERS