Jessica Wang
Chapter 9: Sampling Distributions and Confidence Intervals for
Proportions
Statistics:
o
¯
x
, mean of sample
os, standard deviation of the sample
o
^p
, sample proportion
Parameters:
oµ, mean of population
oσ, standard deviation of population
op, population proportion
Sampling distribution: distribution of the statistic obtained from repeated
samples (or repeated trials of an experiment) using the same number of
observations
oChanges with parameter
Sample proportions: The proportion of “successes” can be more informative
than the count. In statistical sampling the sample proportion of successes is used
to estimate the proportion p of successes in a population.
oFor any SRS of size n, the sample proportion of successes is:
^p=count of successes in the sample
n=X
n
oRules for Sample Proportion:
A population with a fixed proportion p
Random Sample (independent, equal chance)
Sample size is large, np > 9 and n(1-p) > 9
oIf the sample size is much smaller than the size of a population with
proportion p of successes, then the mean and standard deviation of
^p
are:
μ
^p
=p σ
^p
=
√
p(1−p)
n
oBecause the mean is , we say that the sample proportion in an SRS is an
unbiased estimator of the population proportion p
oThe variability decreases as the sample size increases. So larger samples
usually give closer estimates of the population proportion p
Confidence Interval of a Sample Proportion
oStatement: Ex - We are 95% confident that between 24%-2*1.9% and
24%+2*1.9% of people in Washington agree with the recent changes to
bankruptcy laws. < The proportion of the population
o95% confidence 95% of samples of this size will produce confidence
intervals that capture the true proportion of the population (and we
expect 5% of our samples to produce intervals that fail to capture the true
proportion)
