Introductory Biostatistics

In a typical situation, the null hypothesis of a statistical test is concerned
with a parameter; the parameter in this case, with continuous data, is the mean
m. Sample data are summarized into a statistic that is used to estimate the
parameter under investigation. Since the parameter under investigation is the
population meanm, our focus in this case is the sample meanx. In general, a
statistic is itself a variable with a specific sampling distribution (in the context
of repeated sampling). Our statistic in this case is the sample meanx; the cor-
responding sampling distribution is obtained easily by invoking thecentral limit
theorem. With large sample size and assuming that the null hypothesisH 0 is
true, it is the normal distribution with mean and variance given by

mx¼m 0

sx^2 ¼

s^2

n

respectively. The extra needed parameter, the population variances^2 , has to be
estimated from our data by the sample variances^2. From this sampling distri-
bution, the observed value of the sample mean can be converted to standard
units: the number of standard errors away from the hypothesized value ofm 0.
In other words, to perform a test of significance forH 0 , we proceed with the
following steps:

1. Decide whether a one- or a two-sided test is appropriate; this decision
   depends on the research question.


2. Choose a level of significance; a common choice is 0.05.


3. Calculate thetstatistic:

t¼

x
m 0

SEðxÞ

¼

x
m 0

s=

ffiffiffi

n

p

4. From the table fortdistribution (Appendix C) withðn
   1 Þdegrees of
   freedom and the choice ofa(e.g.,a¼ 0 :05), the rejection region is deter-
   mined by:
   (a) For a one-sided test, use the column corresponding to an upper tail
   area of 0.05:

ta
tabulated value forHA:m<m 0

tbtabulated value forHA:m>m 0

(b) For a two-sided test orHA:m 0 m 0 , use the column corresponding to

an upper tail area of 0.025:

ONE-SAMPLE PROBLEM WITH CONTINUOUS DATA 247