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 proportion Alcohol 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: n ƥ = (10, 904) (0.228) = 2486 n(1-ƥ) = 10, 904 ( )= 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=0.228 n=10,904 z*= ƥ ± z* √ ƥ (1-ƥ) n ± √ (0.228) (0.772) 10, ± (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 p Confidence 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