1 / 5

COMPARING PROPORTIONS IN LARGE SAMPLES

Learn how to compare proportions in large samples through a step-by-step statistical hypothesis testing process, with an example of determining if men are more likely to make over $40,000 than women. Understand critical values, test statistics, and decision-making in hypothesis testing.

lindatravis
Download Presentation

COMPARING PROPORTIONS IN LARGE SAMPLES

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. COMPARING PROPORTIONS IN LARGE SAMPLES Examples: Compare probability of H on two coins. Compare proportions of republicans in two cities. 2 populations: p1=proportion of S in population 1, p2=proportion of S in population 2. GOAL: Determine if p1=p2 based on two samples. Perform two Binomial experiments 1ST sample: x successes in m ind. trials, get sample proportion of S: ; 2nd sample: y successes in n ind. trials, get sample proportion of S: . Test Ho: p1= p2 vs Ha: p1≠ p2 or Ha: p1> p2 or Ha: p1< p2

  2. TESTING HYPOTHESIS PROCEDURE Test on significance level α. STEP1.Ho: p1= p2 vs Ha: p1≠ p2 or Ha: p1> p2 or Ha: p1< p2 STEP 2. Test statistic: where, is the “pooled” or combined proportion Under the Ho, the test statistic has standard normal distribution for large samples. STEP 3. Critical value? For one-sided test zα, for two-sided zα/2 . STEP 4. DECISION-critical/rejection region(s) depends on Ha. Ha: p1 ≠ p2 Reject Ho if |z|> zα/2; Ha: p1 > p2 Reject Ho if z > zα; Ha: p1 < p2 Reject Ho if z < - zα. STEP 5. Answer the question in the problem.

  3. EXAMPLE A sample of 180 college graduates was surveyed. 100 of them men and 80 women, and each was asked if they make more or less than $40,000 per year. The following data was obtained. ≥ $40,000 < $40,000 Total Men: 60 40 100 Women: 30 50 100 Total 90 90 180 Are men more likely to make more than $40,000 than women? Use α=0.05. Soln. Let p1= true proportion of men making over $40k; p2= true proportion of women making over $40k;

  4. EXAMPLE, contd. STEP1.Ho: p1= p2 vs Ha: p1> p2 STEP 2. Test statistic: STEP 3. Critical value= zα=z0.05=1.645. STEP 4. DECISION. z = 3 > 1.645, reject Ho. STEP 5. Men are more likely than women to make over $40k.

  5. EXAMPLE contd. Find the p-value for the test P-value = P(Z>z) = P(Z>3) = 0.0013 NOTE. The table on page 194 presents a summary of testing hypotheses formulas for both approaches: critical values and p-values.

More Related