Estimating a Population Estimating a Population ProportionProportionTarget Goal: Target Goal:

I can use normal calculations to I can use normal calculations to construct confidence intervalsconstruct 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.

• Now we will focus on answering questions about the proportion of some outcome of a population.

Population ProportionsPopulation 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

p̂p̂

Ex. Risky Behavior in the Age of AidsEx. 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.

170ˆ 0.0636

2673p

Approximately NormalApproximately 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.

p

• Standard deviation of sample (__________________): Standard Error

p̂ ˆ ˆ1p p

n

StandardizeStandardize

• 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).

p̂

p̂

ˆ

ˆ ˆ

p pz

pqn

• For a confidence interval:

use as an estimate of .

We also replace the standard deviation by the standard error of .

p̂ p

p̂

ˆ ˆ(1 )p pSE

n

Assumptions for Inference about Assumptions for Inference about a Proportion:a Proportion:

1. Random: SRS

2. Independent: Population 10n (when selecting without replacement).

3. Normal:

: *CI estimate z SE

ˆ ˆ10 and 10 np nq

Ex. Risky Behavior cont.Ex. Risky Behavior cont. Are the conditions met? 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.

Step 2:Step 2: Plan – We will use a one-sample Plan – We will use a one-sample z interval for p z interval for p if the conditions are met.if the conditions are met.Does the sample meet the requirements for Does the sample meet the requirements for inference?inference?

1) 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.

2) Independent? population 10n; Yes, overall heterosexual adult population is much larger than 10 times 2673.

3) Normal: for a confidence interval.

2673(0.0636) = 170 ≥ 102673(0.9364) = 2503 ≥ 10

ˆ ˆ10 and 10 np nq

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

Ex. Estimating Risky BehaviorEx. Estimating Risky Behavior

• We are previously given that 170 of 2673 adult heterosexuals had multiple partners.

Compute 99% C.I.

Step 3: DoDiagram:

invnorm(1-.005)

• z* = 2.576 (table A or calc.)

ˆ 0.0636p

Ex. Estimating Risky BehaviorEx. Estimating Risky Behavior

ˆ ˆˆ *

pqp z

n (0.0636)(0.9364)

0.0636 2.5762673

0.0636 0.0122 (0.0514,0.0758)CI

• Step 4: Conclude

We are 99% confident that the actual percent of adult heterosexuals with multiple partners in the past year lies between 5.1% and 7.6%.

• Summary: using Confidence Intervals

Before calculating a confidence interval for µ or p there are three important conditions that you should check. C

on

fiden

ce Inte

rvals: Th

e Ba

sicsC

on

fiden

ce Inte

rvals: Th

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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 Normal if 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.

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