### Chapter 19 * Confidence Intervals for Proportions

```Chapter 19 – Confidence
Intervals for Proportions
Next few chapters
 What percentage of adults own smartphones?
 What is the average SAT score of Baltimore County




students?
Do a higher percentage of women vote for Democrats
than men?
Do cars who use a fuel additive get better fuel
efficiency?
Is it true that 30% of our students work part-time?
Does the average American eat more than 4 meals
out per week?
Confidence Intervals &
Hypothesis Tests
 For the remainder of the semester we are going to
focus on confidence intervals and hypothesis
tests
 Confidence Interval: range of values we predict
the true population statistic is within
 Hypothesis Test: determine whether or not a
claim made about a population statistic is valid
Gallup/Harris Polls
 Gallup:

The percentage of Americans reporting they ate healthy all
day "yesterday" declined to 66.1% in 2011 from 67.7% in
2010
 Nielsen:

Almost half (49.7%) of U.S. mobile subscribers now own
smartphones, as of February 2012.
 Harris:

Currently one in five U.S. adults has at least one tattoo
(21%) which is up from the 16% and 14% who reported
having a tattoo when this question was asked in 2003 and
2008, respectively.
Confidence Intervals
 We use a sample to make our prediction about the
population
 Since each sample we take will give us a slightly
different estimate, we have to understand the
random sampling variation we’ve been studying
 We can never be precise about our estimate, but we
can put it within a range of values we feel confident
Width of Confidence Interval and Confidence
Level
 Sample Size: Should we be more or less confident in
our estimate as sample size increases?
 Confidence Level: Should we expect a wider or
narrower interval as our confidence increases?
 Each interval will have a Margin of Error that takes
all of this into account

Based on sample size and confidence level
Proportion Estimates
 We saw in the last chapter that when we use a
sample to estimate a proportion that the proportion
estimates were distributed Normally with:
( p)  p
SD ( p) 
pq
n
 We know our estimate p is just an estimate, but
want to know how good an estimate it is
Standard Error
 We can use the standard deviation of our sampling
distribution model and our proportion estimate to find
the Standard Error:
SE ( p) 
pq
n
 We can use this error to get a sense for how confident we
are that our estimate is correct
 It is not a mistake we made, but a way to measure the
random sampling variation, and since we don’t have the
population proportion, we can’t know the s.d.
21% of adults have tattoos
 Sample was from 2,016 adults, 423 of which had
tattoos.
 Since this is just one sample, let’s look at the
sampling distribution model like we did last chapter:
model
mean:
SE:
What can we say?
 21% of all adults have tattoos?
 No, this was only 1 sample of 2,016 people
 It’s likely that 21% of all adults have tattoos?
 No, again, with only 1 sample, we’re pretty sure this isn’t the
actual proportion
 While we can’t be sure of the actual proportion of
adults with tattoos, we’re sure it’s between and 19.2%
and 22.8%

We can’t know for sure what the actual proportion is, but this
at least shows some of the uncertainty we have
What we can really say
 We’re pretty sure that the actual proportion of adults
that have tattoos is contained in the interval from
19.2% and 22.8%
 We are, in fact, 95% confident that between 19.2%
and 22.8% of adults have tattoos.

95% confidence uses 2 SD’s as in our 68-95-99.7 Rule
 This is a Confidence Interval which we will
usually write in interval notation: (.192, .228)
Example: Legal Music
 A random sample of 168 students were asked about
their digital music library. Overall, out of 117,709
songs, 23.1% were legal. Construct a 95% confidence
interval for the fraction of legal digital music.
What does the Confidence Interval really mean?
 Technically, a 95% confidence interval means that
95% of all samples of the same given size will include
the true population proportion.
This represents confidence
intervals of 20 simulated
samples for the sea fans
infected from the example
in the text.
You can see that most of the
confidence intervals include
the true proportion.
Figure from DeVeaux, Intro to Stats
Certainty vs. Precision
 If you were going to guess someone’s height, would
you be more likely to be right with a wider or smaller
range for your guess?
 The larger the margin of error you have, the more
likely your prediction is to be correct.
 The more precise we want to be, the less confident
we can be that we are correct.
Margin of Error (ME)
 Our 95% confidence interval used: p ± 2 SE( p )
 We can always think of a confidence interval as:
Estimate ± ME
 Margin of Error is based on the level of confidence.
 We used 2 SE for our margin of error based on the 68-95-99.7
rule.
Critical Values
 While 2 is a good estimate for a 95% confidence
interval, using the Normal probability table, we can
see that z*= 1.96 is more accurate.
 What would be the critical value for a 92%
confidence interval?
92% Confidence Interval
4%
92 %
4%
Use Table in
Appendix D to
find appropriate
z-score.
Calculating Margin of Error
 Using our earlier example involving tattoos, what
would the margin of error be for a 92% confidence
interval?
SE ( p) 
(.2 1 )(.7 9 )
2016
 .0 0 9
Now we also know for a 92% confidence interval, we
use z* = 1.75
ME = 1.75(.009) = .016
(ME for 95% CI: .018)
Assumptions/Conditions
 Independence Assumption
 Randomization Condition
 10% Condition
 Sample Size Assumption
 We will need more data as proportion gets closer to 0 or 1
 Success/Failure Condition
One Proportion Z-Interval
 When conditions are met
Confidence Interval = p  z *
pq
n
Make sure you can interpret your confidence
interval.
Confidence Interval Example
 A Gallup poll shows that 62% of Americans would amend
the Constitution to use the popular vote for Presidential
elections instead of the electoral vote. They used a
random sample of 1,005 adults aged 18+
 Verify that the conditions were met.
 Construct a 95% confidence interval.
 Interpret your interval.
 http://www.gallup.com/file/poll/150272/Americans_Po
pular_Vote_Not_Electoral_College_111024%20.pdf
Choosing Sample Size
 As we pick a larger sample, we should expect our
margin of error to go down. Why?
ME  z
*
pq
n
 If we know our desired Margin of Error, we can solve
for n to get our sample size


Always round up to next integer
If we don’t know p then we use p = 0.5 to max error
Sample Size Example
 If we find from a pilot study that 32% of Math 153
students are full-time students, how many students
would we have to sample to estimate the proportion
of Math 153 full-time students to within 7% with
90% confidence?
```