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 n

ow 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 about

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:

We know our estimate is just an estimate, but want to know how good an estimate it is

p

( )p p ( ) pqSD pn

Standard Error

We can use the standard deviation of our sampling distribution model and our proportion estimate to find the Standard Error:

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.

( ) pqSE pn

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.

Figure from DeVeaux, Intro to Stats

This represents confidence intervals of 20 simulatedsamples for the sea fans infected from the examplein the text.

You can see that most of theconfidence intervals includethe true proportion.

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: ± 2 SE( )

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.

p p

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

92 % 4%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?

Now we also know for a 92% confidence interval, we use z* = 1.75

ME = 1.75(.009) = .016 (ME for 95% CI: .018)

(.21)(.79)( ) .0092016

SE p

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 =

Make sure you can interpret your confidence interval.

* pqp zn

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_Popular_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?

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 then we use = 0.5 to max error

* pqME z

n

p p

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?