Inference for Proportions Section

Starter Do dogs who are house pets have higher cholesterol than dogs who live in a research clinic? A clinic measured the cholesterol level in all 23 of its dogs and found a mean level of 174 with s.d. of 44. They also measured 26 house pets brought in to be neutered one week and found a mean of 193 with s.d. of 68. Is this strong evidence that house pets have higher cholesterol than clinic dogs? What is wrong with this study?

Today’s Objectives Students will use a formula to form a confidence interval for a population proportion. Students will use a formula to perform a hypothesis test about a population proportion. California Standard 17.0 Students determine confidence intervals for a simple random sample from a normal distribution of data and determine the sample size required for a desired margin of error.

The Big Picture (so far) Chapter 11: Estimating means –One population: μ = some constant –Two populations: μ 1 = μ 2 Chapter 12: Estimating proportions

The Distribution of Sample Proportions Recall from chapter 9 that if we take many samples of a population proportion that all such samples have a normal distribution. The mean of the sampling distribution is the true population proportion. The standard deviation of the distribution is given by the formula Since the distribution of sample proportions is normal, and we know its standard deviation, we can use z tests, not t tests.

Assumptions Data are an SRS from the population Population size is at least ten times sample size Sample size is large enough that both the expected “yes” and “no” counts are 10 or more: (Notice that we use p, not p-hat)

Assumptions So Far… Procedure One population t for means Two population t for means One population z for proportions Assumptions SRS, Normal Dist SRS, Normal, Independent SRS, large pop, 10 succ/fail

But we don’t know p. Now what? Hypothesis Tests We start by assuming some value p o Since we assume p o is true, use it in the standard deviation formula as is: Now find the z statistic as usual

Hypothesis Test Example A coin is flipped 4040 times and heads comes up 2048 times. Is this good evidence that the coin is not fair? 1.Are the three assumptions met? 2.State the null and alternative hypotheses 3.What is p-hat? 4.Find the z statistic and associated p-value; draw a conclusion

Assumptions The flips are an SRS of all possible flips. The population (all possible flips) is more than 10 times the sample size. Assuming p =.5, we expect.5 x 4040 = 2020 heads and 2020 tails. Both are greater than 10, so assumption is met. H o : p =.5H a : p ≠.5 (Why 2-sided?) p-hat is 2048 / 4040 =.5069

Calculations normalcdf(.8771,999) =.19 Because H a is 2-tailed, p =.19 x 2 =.38 There is not sufficient evidence (p =.38) to support a claim that the coin is unfair.

Confidence Intervals We don’t know p, so replace standard deviation with standard error (SE). Use the same formula, but replace p with p-hat: Form a C.I. as usual: estimate ± z*SE

Confidence Interval Example A national AIDS survey found that 170 of 2673 adult heterosexuals had multiple partners. Does this meet the three assumptions needed for inference? Form a 99% C.I. for the true proportion of adult heterosexuals with multiple partners.

Assumptions The actual survey design was a complex stratified sample. The result was close to an SRS and may be used. The number of adult heterosexuals is much larger than 10 times the sample size. The counts of “yes” and “no” are much larger than 10: –2673 x.0636 = 170(Or: more than 10 yes) –2673 x.9364 = 2503(Or: more than 10 no)

Calculations.0636 ±.0122 = (.0514,.0758) So we are 99% confident that the true proportion of adult heterosexuals who have multiple partners is between 5.1% and 7.6%.

Today’s Objectives Students will use a formula to form a confidence interval for a population proportion. Students will use a formula to perform a hypothesis test about a population proportion. California Standard 17.0 Students determine confidence intervals for a simple random sample from a normal distribution of data and determine the sample size required for a desired margin of error.

Homework Read pages 660 – 668 Do problems 5-8