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# Section7.4revised.doc

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Western University

Statistical Sciences

Statistical Sciences 2035

Steve Kopp

Fall

Description

Confidence Intervals for A Population Proportion (section 7.4)
Studies often examine categorical data
• categorical data consists of counts (x) or percents (proportions, p)
obtained from counts
• this section presents CI’s for the unknown proportion of success, p, of a
population
Recall: Section 6.2
Select a random sample of size n from a population and record the count, x,
of “successes”
x ~ Bin( n, p)
where p = probability of success We are interested in estimating p, as it is
often unknown
To do this, we use our sample and
calculate the sample proportion of
success
x
p = n
This is our point estimate of p
We know that p will not be equal to p
Instead it will be best to give a confidence interval for which we are
confident that the value of p lies in Recall: From Section 6.2
Sampling Distribution of p
1. Centre: μ ˆp= p
p(1− p)
2. Spread: σ p =
n
3. Shape: Approximately normal if n is “large”
• to see if this normal assumption is valid, you must check:
np ≥ 5 and n(1− p) ≥ 5
• if the above is true, then we can use the normal distribution in our CIs
Note
If the sample size is small, alternative methods must be used Confidence Intervals
A 100(1 − α)% confidence interval for p is
p p(1− p)
± z α/2 n
However, this requires us to know the value of p, which we do not;
Instead we use the value of our sample proportion, p , in its place:
Thus, a 100(1 − α)% CI for p is
p(1− p)
p ± z α/2
n Example 7.7
You wish to estimate the proportion of students at your university who have
jobs where they work for 10 or more hours per week while classes are
session.
We will use our survey of the students in this Statistics 2035 as our random
sample.
The survey collected information from 191 students and 30 of them said that
they work 10 or more hours a week.
Calculate and interpret a 98% confidence interval for the true proportion of
all students at Western who work 10 or more hours a week while classes are
in session. Solution to 7.7 Example 7.8
The operations manager for a large city newspaper wa

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