1. Lecture 6

2. Hypothesis tests for the population mean  Similar arguments to those used to develop the idea of a confidence interval allow us to test the hypothesis that the population mean is equal to a particular value.  For example suppose we were told that the average IQ of UK students was 115, and we were interested in whether or not the mean IQ of MAS1401 students was different from this.

3. Then the hypothesis that µ, the population mean IQ for the MAS1401 students, is equal to 115 is known as the null hypothesis, denoted by H 0. Here the null hypothesis is: H 0 : µ = 115. The other possibility, that the mean IQ of MAS1401 is not 115, is known as the alternative hypothesis, or the experimental hypothesis. Here the alternative hypothesis is: H A : µ ≠ 115.

4.  In order to investigate this we can carry out a hypothesis test.  In the example we are considering, the test we use is called a 1-sample t-test (because there is one sample of data, and we use the t-distribution!)  MINITAB will carry this test out for us, using the procedure “1-sample t...”...  Here, the outcome is p = 0.015. What does this mean?

5.  The outcome of a hypothesis test is always a “p-value”.  The p-value is a measure of probability. It tells us how likely we would be to see a sample as extreme as the one we have actually observed, if, in fact, the null hypothesis were true.  Small p-values tell us that the sample we have observed would be unlikely to have occurred if the null hypothesis really is true. Therefore a small p-value constitutes evidence against the null hypothesis.

6.  For the IQs of the MAS1401 students, we had H 0 : µ = 115, and p = 0.015.  This value of p is a small probability. It tells us that if the mean IQ really is 115 then our sample is very unusual.  A much more plausible explanation is that H 0 is in fact false, and the population mean is not 115, but higher than that.  We use guidelines to interpret the result of a hypothesis test…

7. Guidelines for interpreting p-values:  If p > 0.05, we do not reject the null hypothesis at the 5% level. There is no evidence against the null hypothesis.  If p < 0.05, we reject the null hypothesis at the 5% level. There is moderate evidence against the null hypothesis.  If p < 0.01, we reject the null hypothesis at the 1% level. There is strong evidence against the null hypothesis.  If p < 0.001, we reject the null hypothesis at the 0.1% level. There is very strong evidence against the null hypothesis.  Note that these are guidelines only, and should not be interpreted as hard and fast rules when making decisions!

8. Two independent samples  Lets return to the haematocrit data we looked at in Practical 2. There were measurements on 126 women and 61 men.  We would like to use these data to make comparisons between the population haematocrit distributions for females and males.  The method is to construct confidence intervals for the difference between the mean haematocrit for women and men.

9.  It is also possible to carry out a hypothesis test for whether the difference between the means takes a particular value.  The most commonly tested value is zero, since this amounts to a test of whether or not there is any difference between the population means.  We won’t worry about the details, but we will use Minitab to carry out the work for us, using the 2-sample t-test option “2-sample t...”