April 19th, 2017
Inference One Sample Proportion
Standardized Sample Proportion
Inference about a population proportion p is based on the z statistic that results from
z = (p-hat) — p ÷ √p(1 — p) ÷ n
z has approximately the standard normal distribution as long as the sample is not too small and
the sample is not a large part of the entire population.
Distribution of the Sample Proportion
Because the values of the sample proportion varies from sample to sample, it is a random
variable. So, we have the same questions for the sample proportion as we had for the sample
What is the mean of the sample proportion? p
What is the standard deviation of the sample proportion? SE = √p(1 — p) ÷ n
What is the sampling distribution of the sample proportion if np (1 — p) ≥ 10 or number of
success or failures is 15 or more? (p-hat) ~ N( mean: µ(p-hat) = p, standard error: √p(1 — p) ÷
Large-Sample Conﬁdence Interval for a Proportion
How do we ﬁnd the critical value for our conﬁdence interval?
statistic +/- (critical value) x (standard deviation of statistic)
If the Normal condition is met, we can use a Normal curve. To ﬁnd a level C conﬁdence interval,
we need to catch the central area C under the standard Normal curve.
For example, to ﬁnd a 95% conﬁdence interval, we use a critical value of 2 based on the
68-95-99.7 rule. Using a Standard Normal Table or a calculator, we can get a more accurate
critical value. Note, the critical value z* is actually 1.96 for a 95% conﬁdence level.
Draw an SRS of size n from a population with unknown proportion p of successes. An
approximate level C conﬁdence interval for p is
p +/= z(å/2) √(p-hat) x (1 — p-hat) ÷ n Where z(å/2) is the critical value for the standard Normal density curve with area C between -
z(å/2) and z(å/2). Use this interval only when the counts of suc