Hypothesis Testing For Proportions

1. Hypothesis Testing For Proportions

2. P-Values of a Hypothesis Test • If the null-hypothesis is true, the P-value (probability value) of a hypothesis test is the probability of obtaining a sample statistic with a value as extreme or more extreme than the one determined from the sample data (farther away).

3. And then… • The smaller the P-Value, the more evidence there is to reject the null hypothesis. • A very small P-Value indicates a rare occurrence. • If the P-Value ≤ α, then you will reject the null hypothesis. • If the P-Value > α, then you will fail to reject the null hypothesis.

4. Tests… • If Ha contains <, the test is a left-tailed test. • P is the area to the left of the test statistic. • If Ha contains >, the test is a right-tailed test. • P is the area to the right of the test statistic. • If Ha contains ≠, the test is a two-tailed test. • P is the area to the left of the negative test statistic, and P is the area to the right of the positive test statistic.

5. Definitions • Hypothesis tests for proportions occur (for example) when a politician wants to know the proportion of his or her constituents who favor a certain bill or when a quality assurance engineer tests the proportion of parts which are defective. • Z-Test for a Proportion P: A statistical test for a population proportion P. It can be used when np ≥ 5 and nq ≥ 5. (q is 1-p) • A test statistic is the sample proportion p-hat. The standardized test statistic is z…formula to follow later.

6. Steps to follow for a test… • Verify that np ≥ 5 and nq ≥ 5. If these are true, the distribution for p-hat will be normal and you can continue; otherwise you cannot use normal distribution for the problem. • State the claim…Identify null and alternative hypotheses. • Specify the level of significance (α). • Sketch the sampling distribution (make a curve).

7. More steps… • Determine any critical values (see next slide). These will be borders between rejection regions and non-rejection regions (below). They will be the same values each time. • Determine any rejection regions. These are a range of values for which the Ho is not probable. If a test statistic falls into this region, Ho is rejected. A critical value separates the rejection region from the non-rejection region.

8. Critical Values

9. More steps… • Find the z-score (standard score): • Make a decision to reject or fail to reject Ho. • Interpret the decision in the context of the original claim.

10. Example… • A medical researcher claims that less than 20% of adults in the U.S. are allergic to a medication. In a random sample of 100 adults, 15% say they have such an allergy. At α = 0.01, is there enough evidence to support the researcher’s claim? • n = 100, p = 0.20, q = 0.80 • 1. np = 20, nq = 80…you can continue. • 2. Ho: p ≥ 0.2, Ha: p < 0.2

11. Example (cont’d)… • Since Ha is <, this is a left-tailed test, and since α = 0.01, we will be using the critical value as -2.33 (they use the symbol zo for this). • See drawing on board for sketch. • The rejection region is z < -2.33. • The standardized test statistic (z) is:

12. Example (cont’d)… • Since z = -1.25, and this is not in the rejection region, you should decide not to reject the null hypothesis. • Interpretation: There is not enough evidence to support the claim that less than 20% of adults in the U.S. are allergic to the medication.

13. In your groups… • USA Today reports that 5% of US adults have seen an extraterrestrial being. You decide to test this claim and ask a random sample of 250 U.S. adults whether they have ever seen an extraterrestrial being. Of those surveyed 8% reply yes. At α = 0.01, is there enough evidence to reject the claim? • Your group is to complete and document all the steps to come to the final answer. This will be turned in.

14. Homework… • There will be some on Friday 