Download A 95% confidence interval for a population proportion p is p ˆ

A 95% confidence interval for a population proportion p is
p
p̂ ± 1.96(se), with se = p̂(1 − p̂)/n
where p̂ denotes the sample proportion based on n observations.
Warning: we assume here the sample size is large, i.e. the number
of successes is larger than 15 and the number of failures is larger
than 15.
A 95% confidence interval for the population mean µ is
√
x̄ ± t.025,df (se), with se = s/ n
where t.025,df denotes the t-score of the t-distribution and
df = n − 1 denotes the degrees of freedom of the corresponding
t-distribution.
Effects of Confidence Level and Sample Size on Margin of Error
The margin of error for a confidence interval:
• Increases as the confidence level increases
• Decreases as the sample size increases.
Sample Size for Estimating a Population Proportion
The random sample size n for which a confidence interval for
a population proportion p has margin of error m (such as
m = 0.04) is
p̂(1 − p̂)z 2
n=
m2
The z-score is based on the confidence level, such as z = 1.96
for 95% confidence. We either guess the value for the sample
proportion p̂ based on other information or take the safe
approach of setting p̂ = 0.5.
Sample Size for Estimating a Population Mean
The random sample size n for which a confidence interval for
a population mean µ has margin of error m (such as
m = 0.04) is
σ2z 2
n=
m2
The z-score is based on the confidence level, such as z = 1.96
for 95% confidence. To use this formula, we guess the value
for the population standard deviation σ.