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Comparing Two Proportions

Comparing Two Proportions. A new twist on some old ideas…. When combining proportions we treat proportions and standard errors in much the same way as we did previously when combining sample means Make sure you are aware of the slightly different way in which the formulae appear

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Comparing Two Proportions

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  1. Comparing Two Proportions

  2. A new twist on some old ideas… • When combining proportions we treat proportions and standard errors in much the same way as we did previously when combining sample means • Make sure you are aware of the slightly different way in which the formulae appear • Ultimately, this will usually boil down to a z-score test

  3. Means and Standard Deviations The difference in sample proportions is When n is large, the standard deviation in D is etc!

  4. Confidence testing, intervals, margins etc… • These concepts are still essentially the same • Sometimes a Wilson estimate will be used – note the slight change in formulae • Make sure to grasp the subtlety between standard deviation and standard error in the mean. SE often “creeps in” un-announced.

  5. Examples… Go to Minitab solution • Look at example 8.8 • Are men more frequent binge drinkers? How confident can you be of this result? D = 0.227-0.170 = 0.057 or 5.7% Choose a 95% C: z* = 1.960, SED= 0.00622  m = z* SED The margin of error is 0.012 or 1.2%, so we can conclude that male college students are 5.7% more likely to be binge drinkers than female college students with a margin of error of 1.2%, 19 times out of 20.

  6. How to abuse statistics! • A well meaning senate member of the university gets a hold of the previous stats. She concludes that since 17% of female students and nearly 23% of male students are binge drinkers it means that the ratio of female to male drinkers is 17/23 or 0.75 or 75% This implies that men are 25% more likely to be binge drinkers than women. We should enforce a dry-campus rule. Comment on this conclusion.

  7. Pooled estimates… • An estimate is pooled when it combines the data from two or more data sets • For example – in our analysis of binge drinkers on campus we tacitly assumed that the 5.7% difference between male and female drinkers was significant. How can we justify this?

  8. The incidence of binge drinking could be calculated via: • This is a pooled estimate for the overall incidence of binge drinking since it pooled the results for both groups • The estimate for the standard error takes on a slightly different form: “p” signifies pooled

  9. Relative Risk • Our “zealous” senate member was actually computing a relative risk RR when she expressed the ratio of the two proportions of binge drinkers on campus. • RR is a useful statistic when comparing populations. In our example we could conclude that male college students are about 1.34 times more likely to be binge drinkers than female students. • RR is expressed as

  10. Group Work… • 8.37 • 8.42 • 8.48

  11. In conclusion… • Read the summary on page 595 carefuuly and note the similarity between the ideas developed here and those developed in Chapters 5 and 7 • Be sure to note the role of z-scores in this chapter • Review the new terms – pooled data and relative risk

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