STA2023 Chapter 20 Part 1 Comparing Two Proportions two proportion z test Confidence Intervals for Comparing Two Independent Proportions Parameter was want to estimate is Since we don t know the population parameters we use their estimators Calculator Option STAT TEST Option B 2 PropZInterval Fill in X1 N2 C Level Confidence X2 N1 z invnorm confidence right tail ex invnorm 0 975 if confidence was 95 z 1 64 90 confidence z 1 96 95 confidence z 2 58 99 confidence Interpreting Confidence Intervals for Comparing Two Proportions zero in interval If the confidence interval includes zero therefore evidence of a significant difference between the 2 proportions 0 is plausible There is no statistical If the confidence interval does not include zero and the values in the interval are both positive say a b then is between a and b more than If the confidence interval does not include zero and the values in the interval are both negative say a b then is between a and b more than Assumptions Conditions for Comparing Two Proportions Independent 10 successes and 10 failures in each group 10 condition to make sure we sampled less than 10 of the population Hypothesis Null Hypothesis states that there is no difference between the parameters Alternative Hypothesis Test Statistic z test because it is categorical data But if we only had a and a where did the with no underscore come from Where of successes sample size Calculator Option STAT TEST Option 6 2 PropZTest Fill in X1 N2 P1 P2 P2 P2 X2 N1 P value the probability that the test statistic equals the observed value or a value more extreme if the null hypothesis is true Conclusion small p values support so reject large p values suggest failing to reject Examples 1 Among randomly sampled teens aged 15 17 57 of the 48 boys had social media profiles compared to 70 of the 256 girls Is there a significant difference between the proportion of boys and girls that had social media profiles a Are the assumptions and conditions met to create a confidence interval Independent random samples random one group does not affect another 10 successes 10 failures 179 77 and 141 107 10 condition 504 10 of teen in world b Create a 95 confidence interval and interpret 0 7 0 57 1 96 0 001808619 0 0834 0 13 0 05 0 21 We are 95 confident that the proportion of 15 17 year old girls who use social media is between 0 05 and 0 21 more than the proportion of 15 17 year old boys who use social media c Conduct a two proportion z test to determine if there is a significant difference in the proportion of boys and girls who have social media profiles 179 141 256 248 0 63 0 13 0 00185 3 02 P value P z 3 02 2 0 0025 The probability of observing a sample with a 13 or more difference between boys or girls using social media if there is no difference between boys and girls social media usage 1 confidence interval 1 0 95 0 05 0 0025 0 05 YES We have sufficient evidence at 95 confidence level to We Reject conclude that there is a difference between the proportion of boys and girls with social media profiles 2 A new vaccine was tested to see if it could prevent the ear infections that many infants suffer from Babies about a year old were randomly divided into two groups One group received vaccinations and the other did not The following year 414 of 2459 vaccinated children had ear infections compared to 496 of 2451 unvaccinated children a Check the assumption and conditions and create interpret a 99 confidence interval for the difference in the rates of ear infections Independent random sample ear infection independent children randomly divided 10 successes 10 failures 414 2459 and 496 2451 10 condition less than 10 of all babies in the world STAT TEST B 2 PropZInterval X1 414 N1 2459 X2 496 N2 2451 C Level 0 99 Calculate 0 0625 0 0055 We are 99 confident that the proportion of unvaccinated babies who got ear infections in between 0 0055 and 0 0625 more than the proportion of vaccinated babies b Conduct a hypothesis test to determine if the vaccine helped prevent ear infections Test at the 99 level of confidence 414 496 2459 2451 0 1853 OR STAT TEST 6 2 PropZTest X1 414 N1 2459 X2 496 N2 2451 P1 P2 Calculate z 3 07 p value P z 3 07 0011 The probability of observing a 3 difference between vaccinated and unvaccinated babies who get ear infections if the proportions are the same 1 0 99 0 01 0 0011 0 01 YES At 99 confidence we have sufficient evidence to conclude We Reject that the vaccine worked because less vaccinated children got ear infections than the unvaccinated children