Confidence Interval for p,

Using z Procedure

Conditions for inference about proportion

Center: the mean is ƥ. That is, the sample proportion ƥ is an unbiased estimator of the population proportion p.

Shape: if the sample size is large enough that both np and n(1-p) are at least 10, the distribution of ƥ is approximately Normal.

Spread: ƥ = √ ƥ(1-ƥ)/n provided that the N≤10n

Normality

Independence

Binge Drinking in college estimating a population proportionAlcohol abuse has been described by college

presidents as the number one problem on campus, and its an important cause of death in young adults. How common is it? A 2001 survey of 10, 904 random US college students collected information on drinking behaviour and alcohol-related problems. The researchers defined “frequent binge drinking” as having five or more drinks in a row three or more times in the past two weeks. Acoording to this definition, 2486 students were classified as frequent binge drinkers. That’s 22.8% of the sample. Based on these data, what can we say about the proportion of all college students who have engaged in frequent binge drinking?

… Binge Drinking in College Are the conditions met?

SRS: SRS from 10, 904 US college students.

Normality: the counts of YES and NO responses are much greater than 10:

nƥ = (10, 904) (0.228) = 2486

n(1-ƥ) = 10, 904 (1-0.228)= 8418

Independence: the number of college undergraduates (the population) is much larger than 10 times the sample size n=10 904

Confidence interval for a population proportion

ƥ ± z* √ƥ (1-ƥ)n

Estimating risky behavior Calculating a confidence interval for p

C-level: 99%ƥ=0.228n=10,904z*= 2.576

ƥ ± z* √ƥ (1-ƥ)n

0.228 ± 2.576 √(0.228) (0.772)10,904

0.228 ± 0.010

(0.218, 0.238)We are 99% confident that the proportion of college

undergraduates who engaged in frequent binge drinking lies between 21.8% and 23.8%

Summary in Estimating the population

µ and pConfidence Interval

When population ∂is known (population standard Deviation)

Test statistic:

X ± z* √n

_ _∂

z* σ

√n≤ E

Minimum sample size:

A specific confidence interval formula is correct only under specific conditions.

SRS from the population of interest,

Normality of the sampling distribution,

Independence of observations(N≥10n)

When population ∂is NOT known (population standard Deviation)

Test statistic:

X ± t* √n

_ _s

with: df = n-1

margin of error of a confidence interval gets smaller as

•the confidence level C decreases(z* gets smaller),

•the population standard deviation σ decreases, and

•the sample size n increases.

Estimating the population proportion (p)

p ± z* √p (1-p)n

Test-statistic:

^ ^ ^

E ≥ z* √p (1-p)n

^ ^

Minimum sample size

A specific confidence interval formula is correct only under specific conditions.

SRS from the population of interest,

Normality: Rule of Thumb #2,

Independence Rule of Thumb #1