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
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