Math 160 - CooleyIntro to Statistics OCC
Section 9.3 – P-Value Approach to Hypothesis Testing
Suppose from the previous section, we perform a hypothesis test using the critical value approach. From the test, we can conclude that we can reject the null hypothesis at the 10% significance level, but we cannot reject the null hypothesis at the 5% significance level. This would imply that there is a value smaller than 10%, yet larger than 5% at which we can still reject the null hypothesis. The smallest significance level at which we can reject the null would make the evidence stronger against the null.
For example, suppose that the smallest significance level at which we could reject the null is actually 7%. This means that the null hypothesis can be rejected at any significance level of at least 0.07 and cannot be rejected at any significance level less than 0.07. We have more evidence against the null at a significance level of 0.07 rather than 0.10. Why is this true? Recall from Section 9.1, that the significance level is simply the probability of making a Type I error, that is, of rejecting a true null hypothesis. Thus, the lower the probability, the better.
So, this significance level, which is the lowest possible value at which the null can be rejected is called the
P-value.
P-Value
The P-value of a hypothesis test is the probability of getting sample data at least as inconsistent with the null hypothesis (and supportive of the alternative hypothesis) as the sample data actually obtained. We use the letter P to denote the P-value.
Decision Criterion for a Hypothesis Test Using the P-Value
If the P-value is less than or equal to the specified significance level, reject the null hypothesis; otherwise, do not reject the null hypothesis. In other words, if, reject H0; otherwise, do not reject H0.
P-Value as the Observed Significance Level
The P-value of a hypothesis test equals the smallest significance level at which the null hypothesis can be rejected, that is, the smallest significance level for which the observed sample data results in rejection of H0.
Determining a P-Value
To determine the P-value of a hypothesis test, we assume that the null hypothesis is true and compute the probability of observing a value of the test statistic as extreme as or more extreme than that actually observed. By extreme we mean “far from what we would expect to observe if the null hypothesis is true.
Hypothesis Tests Without Significance Levels: Many researchers do not explicitly refer to significance levels or critical values. Instead, they simply obtain the P-value and use it (or let the reader use it) to assess the strength of the evidence against the null hypothesis.
Guidelines for using the P-value to assess the evidence against the null hypothesis.
P-value / Evidence against H0P > 0.10 / Weak or none
0.05 < P 0.10 / Moderate
0.01 < P 0.05 / Strong
P 0.01 / Very strong
Exercises:
1)The P-value for a hypothesis test is 0.083. For each of the following significance levels, decide whether
the null hypothesis should be rejected.
a)
b)
c)
2)In each part, the P-value has been given for a hypothesis test. For each case, determine the strength of
the evidence against the null hypothesis.
a)
b)
c)
d)
3)The value obtained for the test-statistic, z, in a one-mean z-test is given. The type of test is also given.
Determine the P-value in each case and decide whether at a 5% significance level, the data provide
sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.
a) right-tailed test &
b) left-tailed test &
c) two-tailed test &
1