6.5: Estimating a Population Proportion

Objective: To estimate population proportions under specific conditions with various confidence levels

Warm-Up: Estimating Population Means» A coffee machine dispenses coffee into paper cups. You’re supposed to get 10

ounces of coffee, but the amount varies slightly from cup to cup. Here are the amounts measured in a random sample of 20 cups. Find a 95% confidence interval based off of these sample data to capture the true mean. Is the true mean actually captured in your interval?

Standard Error» The sampling distributions we’ve studied are

Normal.˃ For means

˃ For proportions

ˆ pqSD pn

SD yn

x

Standard Error (continued)» When we don’t know p or σ, we’re stuck, right?

» No. We will use sample statistics to estimate these population parameters.

» Whenever we estimate the standard deviation of a sampling distribution, we call it a standard error.

Standard Error (continued)» For a sample proportion, the standard error is

» For the sample mean, the standard error is

ˆ ˆˆ pqSE pn

sSE yn

x

Confidence Intervals» The sampling distribution model of is centered at

p, with standard deviation .

» Since we don’t know p, we can’t find the true standard deviation of the sampling distribution model, so we need to find the standard error

p̂

pqn

SE( p̂)p̂q̂n

Confidence Intervals (continued)» By the 68-95-99.7% Rule, we know

˃ about 68% of all samples will have ’s within 1 SE of ˃ about 95% of all samples will have ’s within 2 SEs of ˃ bout 99.7% of all samples will have ’s within 3 SEs of

» Consider the 95% level: ˃ There’s a 95% chance that is no more than 2 SEs away from .

˃ So, if we reach out 2 SEs, we are 95% sure that will be in that interval. In other words, if we reach out 2 SEs in either direction of , we can be 95% confident that this interval contains the true parameter proportion.

» This is called a 95% confidence interval.

p̂

p̂p̂

p̂

p̂

Confidence Intervals (continued)

Confidence Intervals (continued)»

What Does 95% Confidence Really Mean?

» Each confidence interval uses a sample statistic to estimate a population parameter.

» But, since samples vary, the statistics we use, and thus the confidence intervals we construct, vary as well.

What Does 95% Confidence Really Mean? (cont.)

» The figure below shows that some of our 95% confidence intervals (from 20 random samples) actually capture the true proportion (the green horizontal line), while others do not:

What Does 95% Confidence Really Mean? (cont.)

» Our confidence is in the process of constructing the interval, not in any one interval itself.

» Thus, we expect 95% of all 95% confidence intervals to contain the true parameter that they are estimating.

Example:

»

Margin of Error» We can claim, with 95% confidence, that the

interval contains the true population mean. ˃ The extent of the interval on either side of is

called the margin of error (ME).

» In general, confidence intervals have the form estimate ± ME.

» The more confident we want to be, the larger our ME needs to be, making the interval wider.

p̂2SE( p̂)

Margin of Error

Margin of Error: Certainty vs. PrecisionRECALL:» To be more confident, we wind up being less

precise. ˃ We need more values in our confidence interval to be more certain.

» Because of this, every confidence interval is a balance between certainty and precision.

» The tension between certainty and precision is always there.

˃ Fortunately, in most cases we can be both sufficiently certain and sufficiently precise to make useful statements.

Margin of Error: Certainty vs. Precision (cont.)» The choice of confidence level is somewhat

arbitrary, but keep in mind this tension between certainty and precision when selecting your confidence level.

» The most commonly chosen confidence levels are 90%, 95%, and 99% (but any percentage can be used).

Critical Values» The ‘2’ in (our 95% confidence interval)

came from the 68-95-99.7% Rule.

» Using a table or technology, we find that a more exact value for our 95% confidence interval is 1.96 instead of 2. ˃ We call 1.96 the critical value and denote it z*.˃ We will use z* critical values for proportions, because it is safe to use

the Normal model for proportions.˃ Proportions have a link between the proportion value and the standard

deviation of the sample proportion, while means do not (which is why we used a t-model to make a better estimate).

» For any confidence level, we can find the corresponding critical value (the number of SEs that corresponds to our confidence interval level).

2 ( )ˆ ˆp SE p

Critical Values (continued)» Example: For a 90% confidence interval, the critical

value is 1.645:

ExampleRecall the Fox News example of a poll of 900 registered voters. They found that 82% of the respondents believed global warming exists. Fox reported a 95% confidence interval with a margin of error of 3%. Using the critical value of z (z*) and the standard error based on the observed proportion, what would be the margin of error for a 90% confidence interval? What’s good and bad about this change?

Example (continued)Think some more about the 95% confidence interval Fox News created for the proportion of registered voters who believe that global warming exists.

1. If Fox wanted to be 98% confident, would their confidence interval be wider or narrower?

2. Fox’s margin of error was about 3%. If they reduced it to 2%, would their level of confidence be higher or lower?

3. If Fox News had polled more people, would the interval’s margin of error have been larger or smaller?

Conditions and Assumptions (Important) » All statistical models are made upon assumptions.

˃ Different models make different assumptions.

˃ If those assumptions are not true, the model might be inappropriate and our conclusions based on it may be wrong.

+ Think about the GPA lab we completed…

» You can never be sure that an assumption is true, but you can often decide whether an assumption is plausible by checking a related condition.

Conditions and Assumptions (cont.)» Here are the assumptions and the corresponding conditions you

must check before creating a confidence interval for a proportion:

» Independence Assumption: We first need to Think about whether the Independence Assumption is plausible. It’s not one you can check by looking at the data. Instead, we check two conditions to decide whether independence is reasonable.˃ Randomization Condition: Were the data sampled at random

or generated from a properly randomized experiment? Proper randomization can help ensure independence.

˃ 10% Condition: Is the sample size no more than 10% of the population?

Conditions and Assumptions (cont.) Sample Size/Normal Assumption: The sample needs to be

large enough for us to be able to use the CLT.

˃ Success/Failure Condition: We must expect at least 10 “successes” and at least 10 “failures.”

+ In other words…

np ≥ 10 and nq ≥ 10

One-Proportion z-Interval» Next we know we state the name of the test.

» When the conditions are met, we are ready to find the confidence interval for the population proportion, p.

» The confidence interval is

where

» The critical value, z*, depends on the particular confidence level, C, that you specify.

p̂z SE p̂

SE( p̂)p̂q̂n

One-Proportion z-Interval» Yes. You can use the calculator again! However, you must

show the formula and your plug-in.

» STAT TESTS» 1-PropZInt» Enter the # of observed successes (x) out of the

sample size (n)» Choose your level of confidence» CALCULATE!

Summary: Steps Finding z-Intervals1. Check Conditions and show that you have checked these!

˃ Random Sample: Can we assume this?˃ 10% Condition: Do you believe that your sample size is less than 10% of the

population size?˃ Success/Failure: We must expect at least 10 “successes” and at least 10

“failures.”np ≥ 10 and nq ≥ 10

2. State the test you are about to conduct (this will come in hand when we learn various intervals and inference tests)˃ Ex) One proportion z-interval

3. Show your calculations for your z-interval

4. Report your findings. Write a sentence explaining what you found.˃ EX) “We are 95% confident that the true mean proportion of men is between

who study physics in college is between 17.3% and 22.6%

Confidence Interval Example» An experiment finds that 27% of 53 subjects report improvement after

using a new medicine. ˃ Create a 95% confidence interval for the actual cure rate. Use z* = 1.96.

˃ Why is this interval so wide?

Confidence Interval Example (continued)» An experiment finds that 27% of 53 subjects report improvement after

using a new medicine. ˃ Make it narrower – 90% confidence.

˃ What are the advantages and disadvantages?

Choosing Your Sample Size» The question of how large a sample to take is an important

step in planning any study, as we mentioned when finding sample sizes in sample mean scenarios.

1) Choose a Margin or Error (ME) and a Confidence Interval Level.

2) The formula requires which we don’t have yet because we have not taken the sample. If you are not using sample data from a pilot study, estimate and account for extra variability. A good estimate for , which will yield the largest value for (and therefore for n) is 0.50.

3) Solve the formula for n. ME z*p̂q̂n

Choosing Your Sample Size Example

A credit card company is about to send out a mailing to test the market for a new credit card. From that sample, they want to estimate the true proportion of people who will sign up for the card nationwide. A pilot study suggest that about 0.5% of the people receiving the offer will accept it. To be within a tenth of a percentage point (0.001) of the true rate with 95% confidence, how big does the test mailing have to be?

Assignment

AP Book Work» Please find the pages and book numbers using the CHS Statistics

online calendar.