Hypothesis Testing for Proportions

The Two Hypotheses Null Hypothesis: Ho This hypothesis is the hypothesis you will “reject” or “fail to reject”. When written as a mathematical statement, it will have an =, ≤, or ≥ as it is a “statement of equality”. The Two Hypotheses Alternative Hypothesis (Ha) This hypothesis is known as a “statement of inequality” as it will have a ≠, <, or > in a mathematical statement. While it is the Null Hypothesis you will “reject” or “fail to reject”, either hypothesis can be the “claim” in the experiment.

The 3 Ways to Write the Null and Alternative Hypotheses 1. Ho: X ≤ value Ha: X > value (this will be right tailed) 2. Ho: X ≥ value Ha: X < value (this will be left tailed) 3. Ho: X = value Ha: X ≠ value (this will be two tailed) The 3 Ways to Write the Null and Alternative Hypotheses

In order to test a hypothesis, you will need to go through a 9 step procedure…

Before starting, you will need the values of n, p, and q. 1. Find the values of np and nq. Both must be greater than or equal to 5 in order to continue (otherwise it’s not a normal distribution). 2. Identify the claim, null hypothesis, and alternative hypothesis. 3. Specify the level of significance (alpha) – this will be given to you. Before starting, you will need the values of n, p, and q. -------------------------- n is the sample size p is the population parameter (this will be in the problem) q is the value 1 - p

4. Sketch the curve. 5. Determine any critical values (table coming up). This will be part of your curve in the form of borders. 6. Determine the rejection region(s)… this is where the answer is not probable. You shade the rejection region(s).

Critical Values Tailed Significance Level Critical Value Left 0.10 -1.28 Right 1.28 Two ±1.645 0.05 -1.645 1.645 ±1.96 0.01 -2.33 2.33 ±2.575 Critical Values

Find the z-score (standard score). Where is it on the curve?: Make a decision to “reject” or “fail to reject” Ho. Interpret the decision in the context of the original claim.

A medical researcher claims that less than 20% of adults in the U. S 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

Since Ha is <, this is a left-tailed test, and since α = 0 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:

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.