Fact Sheet – Two Proportion Test (11.1) This test is used to compare the proportion of one population that has a certain trait to the proportion of a second population that has a certain trait. Example: The proportion of ARC students who own a smart phone is the different than the proportion of students at Sacramento City College that own a smart phone. In a random sample of 103 ARC students, 85 had a smart phone. In a random sample of 98 SCC students, 76 had a smart phone. Test the claim at the 5% significance level. Conditions To test hypotheses regarding two population proportions, p1 and p2, the following three conditions must be met. • The two samples are independently obtained using simple random sampling or through a randomized experiment. •
𝑛" 𝑝" 1 − 𝑝1 ≥ 10 and 𝑛( 𝑝( 1 − 𝑝2 ≥ 10
•
20n1 £ N1 and 20n2 £ N2
Hypothesis Test Step 1 -­‐ Hypotheses You must identify which population will be population 1 and which population will be population 2. In our example, we might designate population 1 to represent ARC students and population 2 to represent Sac City Students. The null hypothesis for a 2-­‐proportion test will always be: 𝐻+ : 𝑝" = 𝑝( which could also be written as: 𝐻+ : 𝑝" − 𝑝( = 0 The alternative hypothesis for a 2-­‐proportion test will be one of the following: 1. 𝐻" : 𝑝" < 𝑝( which could also be written as: 𝐻" : 𝑝" − 𝑝( < 0 2. 𝐻" : 𝑝" > 𝑝( which could also be written as: 𝐻" : 𝑝" − 𝑝( > 0 3. 𝐻" : 𝑝" ≠ 𝑝( which could also be written as: 𝐻" : 𝑝" − 𝑝( ≠ 0 Step 2 – Significance Level 𝛼 should be given in the problem Step 3 – Test Statistics The test statistics for a 2-­‐proportion test is calculated using the following formula: 𝑧=
45 647
4 "64
85 987
85 ∙87
, where 𝑝 =
;5 <;7
=5 <=7
Just write “Two Proportion Test”, rather than writing the test statistic. Developed by George Woodbury in conjunction with the textbook Interactive Statistics: Informed Decisions Using Data, by Michael Sullivan III & George Woodbury, 2016. Modified with permission by Sharleen McCarroll. Step 4 – Calculate the P-­‐value with StatCrunch StatCrunch Directions – 2-­‐Proportion Hypothesis Test • Stat > Proportion > Two Sample > With Summary • Enter the number of Success and Trials for both samples • Under Perform: Choose Hypothesis Test. o Leave: 𝐻+ : 𝑝" − 𝑝( = 0 o For the alternative, 𝐻> , choose the correct symbol <, >, 𝑜𝑟 ≠ • Click Compute! Step 5 – Decision about Ho, Conclusion about H1 If the p-­‐value is less than a, reject H0. This means that there is sufficient evidence to conclude that “H1 is true”. (You must write out “H1 is true” in terms of the actual problem. DO NOT JUST WRITE “H1 is true” ) If the p-­‐value is NOT less than a, fail to reject H0. This means that there is NOT sufficient evidence to conclude that “H1 is true”. (You must write out “H1 is true” in terms of the actual problem.) Confidence Interval for the Difference of 2 proportions StatCrunch Directions – 2-­‐Proportion Confidence Interval • Stat > Proportion > Two Sample > With Summary • Enter the number of Success and Trials for both samples • Under Perform: Choose Confidence Interval. o Level: Insert confidence level • Click Compute!