Basic Statistics

100 TESTS OF HYPOTHESES ON POPULATION MEANS

acyanotic congenital heart disease children learn to walk at an older age than normal
children. The entire rejection region is in the upper tail of the normal distribution.
For a: = .05, z[.95] = 1.65, and the null hypothesis is rejected if z is > 1.65.
Sometimes we only wish to reject the hypothesis if the population mean is too
small. For example, suppose that we are testing the weight of a medication; the
producers say that the bottles hold medication weighing 6 oz. We would only want
to reject their claim if the bottles weigh too little. Here, the entire rejection region
is in the lower tail. We would state our null hypothesis as Ho : p 2 6 and reject the
null hypothesis if z 5 -1.65 if we wish to test at an cv = .05 level. Note that for
z[-1.65] we have by symmetry 5% of the area in the lower or left tail of the normal
distribution.

8.1.3 Summary of Procedures for Test of Hypotheses

The usual steps that are taken in testing a null hypothesis include:

1. State the purpose of making the test. What question is to be answered?


2. State the null hypothesis to be tested (two-sided or one-sided).


3. Choose a level of cv that reflects the seriousness of deciding that the null hy-
   pothesis is false when actually it is true.


4. Decide on the appropriate test statistic to use for testing the hypothesis. Here,
   since we are assuming that o is known, the test statistic is z = (x - po)/oz.
   Note that the o is usually not known, so the test statistics given in the next two
   sections are used.


5. Check that the sample and data meet the assumptions for the test. In this
   case we are assuming a simple random sample taken from a population that
   has o equal to the assumed value. We are assuming that z follows a normal
   distribution. We should check that there are no errors or outliers in our sample
   and that the distribution is sufficiently close to normal that the sample mean is
   approximately normally distributed.


6. Compute the value of the test statistic, in this case


7. Either find the tabled values for a: and check the computed test statistic against
   the tabled values or obtain the P value for the numerical value of the test
   statistic. For example, in this case for a two-sided test and a: = .05, see if
   the computed z is > 1.96 or < -1.96. Otherwise, take the computed z value
   and look up the area in Table A.2 or in a computer program. Double the area
   found in one tail to obtain the P value for a two-sided test. For most tests, if
   the standard deviation is not known, the results can be obtained from computer
   programs.