Chapter 9 Testing the Difference Between Two Proportions

What Will I Learn in Ch 9. 6 Objectives: • Test the difference between two proportions

5 -Step Process/Z-Test for Difference Between means We will use the same 5 -Step process but with a couple of differences. • Step 0 – Assumptions: (assumptions are slightly different) • Step 1 - State the Hypothesis (Slightly Different) instead of μ 1 and μ 2 we use p 1 and p 2. • Step 2 - Critical Value(s) (The Same) • Step 3 - Test Value (Different) • Step 4 - Make the decision (The Same) • Step 5 – Summarize the results (The Same)

Testing the difference between 2 proportions: z-test for difference of proportions

What do the hypotheses look like? • If the researcher’s claim is that there is a difference between two proportions, the null and alternative hypothesis will look like

Test value for Testing the Difference Between Proportions

Example: Vaccination Rates In the nursing home study mentioned in the chapter-opening Statistics Today, the researchers found that 12 out of 34 small nursing homes had a resident vaccination rate of less than 80%, while 17 out of 24 large nursing homes had a vaccination rate of less than 80%. At α = 0. 05, test the claim that there is no difference in the proportions of the small and large nursing homes with a resident vaccination rate of less than 80%. Step 0 – Assumptions: Step 1: State the hypotheses and identify the claim. H 0: p 1 – p 2 = 0 (claim) and H 1: p 1 – p 2 0

Example: Vaccination Rates In the nursing home study mentioned in the chapter-opening Statistics Today, the researchers found that 12 out of 34 small nursing homes had a resident vaccination rate of less than 80%, while 17 out of 24 large nursing homes had a vaccination rate of less than 80%. At α = 0. 05, test the claim that there is no difference in the proportions of the small and large nursing homes with a resident vaccination rate of less than 80%. Two-Tailed Test = α/2 =. 05/2 =. 025 Step 2: Find the critical value. Since α = 0. 05, the critical values are z= – 1. 96 and z=1. 96.

Example: Vaccination Rates In the nursing home study mentioned in the chapter-opening Statistics Today, the researchers found that 12 out of 34 small nursing homes had a resident vaccination rate of less than 80%, while 17 out of 24 large nursing homes had a vaccination rate of less than 80%. At α = 0. 05, test the claim that there is no difference in the proportions of the small and large nursing homes with a resident vaccination rate of less than 80%. Step 3: Compute the test value.

Using Geo. Gebra to compute the test value Step 3: Compute the test value. In the nursing home study mentioned in the chapter-opening Statistics Today, the researchers found that 12 out of 34 small nursing homes had a resident vaccination rate of less than 80%, while 17 out of 24 large nursing homes had a vaccination rate of less than 80%. At α = 0. 05, test the claim that there is no difference in the proportions of the small and large nursing homes with a resident vaccination rate of less than 80%.

Step 4: Make the decision. Reject the null hypothesis. Step 5: Summarize the results. There is enough evidence to reject the claim that there is no difference in the proportions of small and large nursing homes with a resident vaccination rate of less than 80%.

Example: Texting While Driving A survey of 1000 drivers this year showed that 29% of the people send text messages while driving. Last year a survey of 1000 drivers showed that 17% of those send text messages while driving. At α = 0. 01, can it be concluded that there has been an increase in the number of drivers who text while driving? Step 0 – Assumptions: Step 1: State the hypotheses and identify the claim. H 0: p 1 = p 2 and H 1: p 1 > p 2 (claim) Step 2: Find the critical value. Since α = 0. 01, the critical value is z = 2. 33.

A survey of 1000 drivers this year showed that 29% of the people send text messages while driving. Last year a survey of 1000 drivers showed that 17% of those send text messages while driving. At α = 0. 01, can it be concluded that there has been an increase in the number of drivers who text while driving? Step 3: Compute the test value.

Step 4: Make the decision. Reject the null hypothesis since z=6. 38 is in the critical region. Step 5: Summarize the results. There is enough evidence to support the claim that the proportion of drivers who send text messages is larger this year than it was last year.