Confidence Intervals for Proportions

Presentation 9.1

Confidence Interval for p

• Remember the purpose of a confidence interval.

• We are simply trying to estimate the true value of the proportion p in the population.

• The confidence interval provides a range of values for p in the population that could reasonably have produced the sample we observed.

Level of Confidence

• Confidence Intervals include a statement of a confidence level, typically 95%.

• You should know how to compute confidence intervals for any level of confidence, but particularly for 80%, 90%, 95%, 98%, 99%.

• The formula is the same for each, but the critical value z* changes.

Level of Confidence

• The level of confidence refers to the reliability of the confidence interval to produce intervals that contain the true p.

• Why not do a 100% confidence interval?– Then we would be completely sure that the

interval has contained the true p.

The 100 % Confidence Interval

• A 100% confidence interval for p is (0,1).– This interval is guaranteed to contain p. – This interval is not very useful as it tells us nothing.

• It is like saying the proportion of people in Spokane who can read is between 0% and 100%.

• This illustrates the trade-off between level of confidence and the usefulness of an interval.

• 90, 95, or 99 percent confidence levels are the most typical.

Confidence Interval Formula• A confidence interval for the population p is given

by:

n

ppZp

)ˆ1(ˆˆ *

Notice the p-hat is used in the standard error. That is because we do not know what p is (remember our purpose is to estimate p).

Sample proportion Critical value

(depends on confidence level)

Standard Error

The Critical Value z*

• For any confidence level, z* is obtained by one of two methods:– Method 1: Look up z* in Table B of your formula packet.

• Remember that the critical values of z are like looking up t with an infinite sample size.

• Use the last row of Table B.• This is the easiest method.

– Method 2: Sketching a standard normal curve and then using invNorm to find z*.

• For a 95% interval, shade the middle 95% of the curve (see picture on the next slide).

• That means there is 2.5% above and 2.5% below the shading.• So, Use invNorm(.975) since 97.5% of the area is to the left of the

positive z*.• This gives z*=1.96.

Finding Z*

Common Critical Values

Confidence level

z critical value

80% 1.2890% 1.64595% 1.9698% 2.3399% 2.5899.8% 3.0999.9% 3.29

The condensed table at the right displays the most commonly used z* values.

Example #1

• A new treatment for fleas. In an experiment, 85% of 200 dogs were rid of their fleas after 3 days of the treatment. What is the reasonable range for the cure rate p of our new treatment? Construct a 95% confidence interval.

)8994,.8006(.

0494.85.

)0252(.96.185.200

)85.1(85.96.185.

Example #1

• The reasonable range for the true proportion is (.8006,.8994).

• That is, there is a 95% chance that our interval caught the true proportion.

M&Ms Example #2

• What is the proportion of yellow candies in a bag of m&ms?

• Let’s take a sample to try to determine this.

• Let’s say a bag of m&ms represents a random sample of size n from the population of these candies.

M&Ms Example #2

• In a 1.69 ounce bag of m&ms you count 7 yellows among the 56 total candies.

• Construct the 95% confidence interval. )2116,.0384(.

0866.125.

)0442(.96.1125.56

)125.1(125.96.1125.

M&Ms Example #2

• Based on our sample, there are somewhere between about 4% and 21% yellows in a bag of m&ms.

What Does 95% Confidence Mean Anyway?

• A 95% confidence interval means that the method used to construct the interval will produce intervals containing the true p in about 95% of the intervals constructed.

• This means that if the 95% CI method was used in 100 different samples, we would expect that about 95 of the intervals would contain the true p, and about 5 intervals would not contain the true p.

Diagram of Confidence

p

95% of intervalsContain true p, butSome do not. About5% miss truth.

This interval missed p.

CI BehaviorYou can manipulate the width of the interval by: - changing the sample size

- changing the confidence level

Confidence Interval Meaning

• We never know if our confidence interval has contained the true p or not, but we know the method we used has the property that it catches the truth 90% of the time (for a 90% level of confidence).

Cautions !• Don’t suggest that the true proportion (parameter)

varies:– There is a 95% chance the true proportion of yellow

m&ms is between .04 and .21.– This kind of sounds like the true proportion is wandering

around in the interval.– Remember the true proportion is either in the interval or

NOT in the interval.• Don’t claim that other samples will agree with

yours.– A different sample will yield a different confidence

interval.– 95% of all possible samples will create intervals (most

different) that will capture the true proportion.

Cautions!

• Don’t be certain about the true proportion (parameter).– The proportion of yellows is between 4 and 21 percent.– This makes it seem like the true p could never be outside this

range. We are not sure of this, just 95% sure.• Remember the confidence interval describes the

parameter (not the statistic).– Never, ever say that we are 95% sure the sample proportion is

between .04 and .21.– We absolutely know for sure that the sample proportion is in

there; we centered the interval around the sample proportion!• Be sure the sample represents the population and that

we don’t generalize our findings beyond what our sample represents.

Confidence Intervals for Proportions

• This concludes the presentation.