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8.2a h.w: pg 496: 35, 37, 41, 43, 47

Estimating a Population Proportion Target Goal: I can use normal calculations to construct confidence intervals. 8.2a h.w: pg 496: 35, 37, 41, 43, 47. Up to this point we have been making inferences about population means.

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8.2a h.w: pg 496: 35, 37, 41, 43, 47

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  1. Estimating a Population ProportionTarget Goal: I can use normal calculations to construct confidence intervals. 8.2a h.w: pg 496: 35, 37, 41, 43, 47

  2. Up to this point we have been making inferences about population means. • Now we will focus on answering questions about the proportion of some outcome of a population.

  3. Population Proportions • The proportion of a population having a given characteristic is a parameter, p. • The proportion of a sample having a given characteristic is a statistic, - “p-hat”: = count of “successes” in the sample count of observations in the sample

  4. Ex. Risky Behavior in the Age of Aids • National Aids Behavioral survey interview a random sample of 2673 adult heterosexuals. Of these, 170 had more than one partner in the past year.

  5. Approximately Normal • We know that the sampling proportion of p is approximately normal for sufficiently large samples: If np and nq ≥ 10 with mean , then the sampling distribution is normal.

  6. Standard deviation of sample (__________________):

  7. Standardize • To standardize, subtract the mean and divide by the standard deviation. • This gives the z test statistic: • The statistic z has approximately the standard normal distribution N(0,1).

  8. For a confidence interval: use as an estimate of . We also replace the standard deviation by the standard error of .

  9. Assumptions for Inference about a Proportion: 1. Random: SRS 2. Independent: Population  10n (when selecting without replacement). 3. Normal:

  10. Ex. Risky Behavior cont. Are the conditions met? • Step1: State - We want to use the National AIDS Behavioral Surveys data to give a confidence interval for the proportion of adult heterosexuals who have had multiple partners.

  11. Step 2:Plan – We will use a one-sample z interval for p if the conditions are met.Does the sample meet the requirements for inference? • Random: SRS? The sampling design indicated “random sample”. In fact it was a complex stratified sample that used inference procedures. The overall effect was close to a SRS so we assume SRS. • Independent? population  10n; Yes, overall heterosexual adult population is much larger than 10 times 2673.

  12. Normal: for a confidence interval. 2673(0.0636) = 170 ≥ 10 2673(0.9364) = 2503 ≥ 10

  13. First requirement (SRS) is only approximately met. The 2nd and 3rd are easily met.

  14. Ex. Estimating Risky Behavior • We are previously given that 170 of 2673 adult heterosexuals had multiple partners. Compute 99% C.I. Step 3: Do Diagram: invnorm(1-.005) • z* = 2.576 (table A or calc.)

  15. Ex. Estimating Risky Behavior

  16. Step 4: Conclude We are 99% confident that the actualpercent of adult heterosexuals with multiple partners in the past year lies between 5.1% and 7.6%.

  17. Summary: using Confidence Intervals Before calculating a confidence interval for µ or p there are three important conditions that you should check. Confidence Intervals: The Basics 1) Random: The data should come from a well-designed random sample or randomized experiment. • 2) Normal:The sampling distribution of the statistic is approximately Normal. • For means:The sampling distribution is exactly Normal if the population distribution is Normal. When the population distribution is not Normal, then the central limit theorem tells us the sampling distribution will be approximately Normalif n is sufficiently large (n ≥ 30). • For proportions:We can use the Normal approximation to the sampling distribution as long as np ≥ 10 and n(1 – p) ≥ 10. 3) Independent: Individual observations are independent. When sampling without replacement, the sample size n should be no more than 10% of the population size N (the 10% condition) to use our formula for the standard deviation of the statistic.

  18. Read 484 - 490

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