9. Hypothesis Testing
Hypothesis testing for means for sample set sizes in where is used as an estimate for is the same as for except that and not is the test statistic. Specifically, the test statistic is
for from any of the hypotheses listed in the table you saw in the previous section (one- and two-tailed versions):
The critical statistic is found in the t Distribution Table with the degrees of freedom .
Example 9.4 : A physician claims that joggers, maximal volume oxygen uptake is greater than the average of all adults. A sample of 15 joggers has a mean of 40.6 ml/kg and a standard deviation of 6 ml/kg. If the average of all adults is 36.7 ml/kg, is there enough evidence to support the claim at ?
2. Critical statistic.
In the t Distribution Table, find the column for one-tailed test at and the line for degrees of freedom . With that find
3. Test statistic.
To compute this we need : , and from the problem statement. From the hypothesis we have . So
At this point we can estimate the -value using the t Distribution Table, which doesn’t have as much information about the -distribution as the Standard Normal Distribution Table has about the -distribution, so we can only estimate. The procedure is: In the row, look for values that bracket . They are 2.145 (with in the column heading for one-tailed tests) and 2.624 (associated with a one-tail ).
is our estimate for .
Reject . We can also base this decision on our -value estimate since :
There is enough evidence to support the claim that the joggers’ maximal volume oxygen uptake is greater than 36.7 ml/kg using a -test at .
Fine point. When we use in a (or test) as an estimate for , we are actually assuming that distribution of sample means is normal. The central limit theorem tells us that the distribution of sample means is approximately normal so generally we don’t worry about this restriction. If the population is normal then the distribution of sample means will be exactly normal. Some stats texts state that we need to assume that the population is normal for a -test to be valid. However, the central limit theorem’s conclusion guarantees that the -test is robust to violations of that assumption. If the population has a very wild distribution then may be bad estimate for because the distribution of sample values will not follow the distribution. The chance if this happening becomes smaller the larger the , again by the central limit theorem.
Origin of the -distribution
We can easily define the -distribution via random variables associated with the following stochastic processes. Let :
Then the random variable
is a random variable that follows a -distribution with degrees of freedom.