Testing the Difference Between Proportions in Nursing Home Vaccination Rates
This content discusses a statistical analysis of vaccination rates in small and large nursing homes to determine if there is a difference in proportions of residents vaccinated. It covers hypothesis testing, critical value calculation, test value computation, decision-making, and result summary.
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Section 9.4 Testing the Difference Between Proportions 1 Bluman, Chapter 9
Test 9 Friday March 14 2 Bluman, Chapter 9
Review of symbols p p X Sample proportion p = n p Population proportion q q
Hypotheses; two tailed H0: p1=p2 OR H0: p1- p2=0 H1: p1 p2 H1: p1 - p2 0
9.4 Testing the Difference Between Proportions ( 1 2 = where ) ( 1 n ) p p p p 1 2 z 1 n + pq 1 2 + + X n X n X n X n = = p p 1 2 1 1 1 2 1 = = p 1 2 q p 2 2 5 Bluman, Chapter 9
Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances Section 9-4 Example 9-9 Page #505 6 Bluman, Chapter 9
Example 9-9: 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 = 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%. 12 0.35 and 34 12 17 29 34 24 58 + + n n 17 24 X n X X n = = = = = = p 0.71 1 2 p 1 2 1 2 + + X = = = = = 0.5, 0.5 1 2 p q 1 2 7 Bluman, Chapter 9
Example 9-9: Vaccination Rates Step 1: State the hypotheses and identify the claim. H0: p1 p2 = 0 (claim) and H1: p1 p2 0 Step 2: Find the critical value. Since = 0.05, the critical values are -1.96 and 1.96. Step 3: Compute the test value. ( 1 = z ) ( 1 n ) ( ) ( ) 1 34 0.35 0.71 p p p p 0 2 1 2 = 1 24 1 n ( )( ) + 0.5 0.5 + pq 1 2 = 2.7 8 Bluman, Chapter 9
Example 9-9: Vaccination Rates 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%. 9 Bluman, Chapter 9
Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances Section 9-4 Example 9-10 Page #506 10 Bluman, Chapter 9
Example 9-10: Missing Work In a sample of 200 workers, 45% said that they missed work because of personal illness. Ten years ago in a sample of 200 workers, 35% said that they missed work because of personal illness. At = 0.01, is there a difference in the proportion? p To compute , you must find X1andX2. ( 1 1 1 0.45 200 = = X p n ( 2 2 2 0.35 200 = = X p n ) = 90 ) = 70 + + + + 90 200 70 200 160 400 X n X n = = = = = 0.4, 0.6 p q 1 2 1 2 11 Bluman, Chapter 9
Example 9-10: Missing Work Step 1: State the hypotheses and identify the claim. H0: p1 = p2 and H1: p1 p2 (claim) Step 2: Find the critical value. Since = 0.01, the critical values are -2.58 and 2.58. Step 3: Compute the test value. ( ) ( 1 2 = = ) ( ) ( ) 1 200 0.45 0.35 p p p p 0 1 2 = z 1 1 n 1 n ( )( ) + 0.4 0.6 + pq 200 1 2 2.04 12 Bluman, Chapter 9
Example 9-10: Missing Work Step 4: Make the decision. Do not reject the null hypothesis. Step 5: Summarize the results. There is not enough evidence to support the claim that there is a difference in proportions. 13 Bluman, Chapter 9
Confidence Interval for the Difference Between Proportions Formula for the confidence interval for the difference between proportions 1 1 p q n p q ( ) + p p 2 n 2 z p p 1 2 2 1 2 1 2 1 1 p q n p q ( ) + + p p z 2 n 2 1 2 2 1 2 14 Bluman, Chapter 9
Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances Section 9-4 Example 9-11 Page #508 15 Bluman, Chapter 9
Example 9-11: Confidence Intervals Find the 95% confidence interval for the difference of the proportions for the data in Example 9 9. 12 34 17 24 X n X n = = = 1 0.35 = = p q 0.35 and 0.65 1 1 1 1 = = = 1 0.71 = = p q 0.71 and 0.29 2 2 2 2 1 1 p q n p q ( ) + p p 2 n 2 z p p 1 2 2 1 2 1 2 1 1 p q n p q ( ) + + p p 2 n 2 z 1 2 2 1 2 16 Bluman, Chapter 9
Example 9-11: Confidence Intervals Find the 95% confidence interval for the difference of the proportions for the data in Example 9 9. ( )( 0.35 0.65 0.35 0.71 1.96 34 ) ( )( 24 ) 0.71 0.29 ( ) + p p 1 2 ( )( 34 ) ( )( 24 ) 0.35 0.65 0.71 0.29 ( ) 0.35 0.71 + + 1.96 0.36 + 0.242 0.36 0.242 p p 1 2 0.602 0.118 p p 1 2 Since 0 is not contained in the interval, the decision is to reject the null hypothesis H0: p1 = p2. 17 Bluman, Chapter 9
On your own Study the examples in section 9.4 Sec 9.4 page 510 #3,7,11,15,17,19 18 Bluman, Chapter 9
