Business Statistics - QBM117 The p value

Objectives Calculation of the p-value of a hypothesis test. Interpretation of the p- value. Testing a hypothesis using the p-value.

What is the p-value of a test? One of the main problems with the testing procedure used so far is that if we change the level of significance, this may change the test’s conclusion. One way of avoiding this is by reporting the p-value of the test. The p-value of a test is the smallest value of  that would lead to rejection of the null hypothesis. It is a measure of the likelihood of obtaining a particular sample mean or sample proportion, if the null hypothesis is true.

How do we calculate the p-value of a test? The p-value for a one tail test is found by for an upper tail test. p-value z

How do we calculate the p-value of a test? The p-value for a one tail test is found by for a lower tail test. p-value z

The p-value for a two tailed test is found by z

Exercise 10.19 p342 (9.19 p308 abridged) p-value z 2.63

Exercise 10.18 p342 (9.18 p308 abridged) 1/2 p-value z -1.76

Interpretation of the p-value The p-value is very important because it measures the amount of statistical evidence that supports the alternative hypothesis. For a given hypothesis test, as the sample statistic moves further away from the hypothesised population parameter, the test statistic gets larger and the p-value gets smaller. ie there is more evidence to indicate that the alternative hypothesis is true.

Using the p-value to test hypotheses A small p-value indicates that if the null hypothesis is true, then the probability of obtaining a sample result as extreme as that obtained is very small ie highly unlikely. Therefore small p-values lead to rejection of the null hypothesis. A large p-value indicates that if the null hypothesis is true, then the probability of obtaining a sample result as extreme as that obtained is reasonably high ie quite likely. Therefore large p-values lead to non-rejection of the null hypothesis.

What is considered to be a small p-value? This depends on the researcher however If the p-value <  it would be considered small and hence the researcher would reject H0 The researcher would then conclude there is sufficient evidence to support the alternative hypothesis. If the p-value >  it would not be considered small and hence the researcher would not reject H0 . The researcher would then conclude there is insufficient evidence to support the alternative hypothesis.

Exercise 10.70 p360 (9.70 p328 abridged) Step 1 Step 2 Step 3

Region of non-rejection 0.95 z α = 0.05 Critical value 1.645 Step 4

Step 5 Step 6 Since 2.9 > 1.645 we reject H0. Since 0.0019 < 0.05 we reject H0. There is sufficient evidence at  = 0.05 to conclude that the campaign was a success.

How do we calculate the p-value of a test when our test statistic is a t statistic? The p-value for a one tail test is found by for an upper tail test. p-value t However the process for calculating this is not so simple.

p-value t The p-value for a one tail test is found by for a lower tail test. p-value t

Example Exercise 10.26 p346 (9.26 p312 abridged) You will remember that the hypotheses we were testing were The test statistic for this test was t = -3.87 t -3.87

Since n = 15, the degrees of freedom here are 14. From Table 4, appendix C we find 0.005 2.977

The p-value we require is Therefore due to the symmetric nature of the t distribution 0.005 -2.977 The p-value we require is Therefore the best we can say is that the p-value < 0.005

Exercises to be completed before next lecture S&S 10.19 10.21 10.25 10.53 10.58 10.67 (9.19 9.21 9.25 9.53 9.58 9.67 abridged)