Chapter 19: Confidence Intervals for ProportionsAP Statistics

Unit 5

Standard Error

0 Both of the sampling distributions we’ve looked at are Normal.0 For proportions

0 For means

ˆ pqSD p

n

SD yn

x

Standard Error (cont.)

0 When we don’t know p or σ, we’re stuck, right?

0 Nope. We will use sample statistics to estimate these population parameters.

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

Standard Error (cont.)

0 For a sample proportion, the standard error is

0 For the sample mean, the standard error is

ˆ ˆˆ

pqSE p

n

sSE y

nx

A Confidence Interval

0Recall that the sampling distribution model of is

centered at p, with standard deviation .

0Since 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:

pq

n

SE( p̂)p̂q̂

n

A Confidence Interval (cont.)

0 By the 68-95-99.7% Rule, we know0 about 68% of all samples will have ’s within 1 SE of p0 about 95% of all samples will have ’s within 2 SEs of

p0 about 99.7% of all samples will have ’s within 3 SEs

of p

0 We can look at this from ’s point of view…

p̂

p̂

p̂

p̂

A Confidence Interval (cont.)

0 Consider the 95% level: 0 There’s a 95% chance that p is no more than 2 SEs

away from .

0 So, if we reach out 2 SEs, we are 95% sure that p 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 proportion.

0 This is called a 95% confidence interval.

p̂

p̂

p̂

A Confidence Interval (cont.)

ExampleA Pew Research study asked cell phone owners if they’ve ever received unsolicited text messages from advertisers. Seventeen percent reported that they had. Pew estimates a 95% confidence interval to be 0.170.04 or between 13% and 21%. Are the following statements about people who have cell phones correct?1. In Pew’s sample, somewhere between 13% and 21% of respondents

reported that they had received unsolicited advertising texts.2. We can be 95% confident that 17% of US cell owners have received

unsolicited advertising texts.3. We are 95% confident that between 13% and 21% of all US cell

owners have received unsolicited advertising texts.4. We know that between 13% and 21% of all US cell owners have

received unsolicited advertising texts.5. 95% of all US cell owners have received unsolicited advertising

texts.

What Does “95% Confidence” Really Mean?

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

0 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.)

0 The figure to the right shows that some of our confidence intervals (from 20 random samples) capture the true proportion (the green horizontal line), while others do not:

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

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

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

Example0  

Margin of Error: Certainty vs. Precision

Margin of Error: Certainty vs. Precision (cont.)

0 To be more confident, we wind up being less precise. 0 We need more values in our confidence interval to be

more certain.

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

0 The tension between certainty and precision is always there.0 Fortunately, in most cases we can be both sufficiently

certain and sufficiently precise to make useful statements.

Margin of Error: Certainty vs. Precision (cont.)

0 The choice of confidence level is somewhat arbitrary, but keep in mind this tension between certainty and precision when selecting your confidence level.

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

Example

0A January 2007 Fox poll of 900 registered voters reported a margin of error of 3%. It is a convention among pollsters to use a 95% confidence level and to report the “worst case” margin of error, based on p = 0.5. How did Fox calculate their margin of error?

Critical Values

0 The ‘2’ in (our 95% confidence interval) came from the 68-95-99.7% Rule.

0 Using a table or technology, we find that a more exact value for our 95% confidence interval is 1.96 instead of 2. 0 We call 1.96 the critical value and denote it z*.

0 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 (cont.)

0 Example: For a 90% confidence interval, the critical value is 1.645:

Example 0 Recall 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 (cont.) 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?

Assumptions and Conditions

0 All statistical models are made upon assumptions.

0 Different models make different assumptions.

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

0 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.

Assumptions and Conditions (cont.)

0 Here are the assumptions and the corresponding conditions you must check before creating a confidence interval for a proportion:

0 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.

0 Randomization Condition: Were the data sampled at random or generated from a properly randomized experiment? Proper randomization can help ensure independence.

0 10% Condition: Is the sample size no more than 10% of the population?

Assumptions and Conditions (cont.)

Sample Size Assumption: The sample needs to be large enough for us to be able to use the CLT.

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

One-Proportion z-Interval

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

0 The confidence interval is

where

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

p̂z SE p̂ SE( p̂)

p̂q̂

n

Graphing Calculator to the Rescue!

0Your graphing calculator will calculate your confidence intervals:0 STAT TESTS0 1-PropZInt0 Enter the # of observed successes (x) out of the sample

size (n) –make sure this is a whole number (use rounding rules—no decimals)!

0 Choose your level of confidence0 CALCULATE!

Summary: Steps Finding One-Proportion z-Intervals

1. Check Conditions and show that you have checked these!0 Random Sample: Can we assume this?0 10% Condition: Do you believe that your sample size is less than

10% of the population size?0 Success/Failure: We must expect at least 10 “successes” and at least

10 “failures.”n ≥ 10 and n ≥ 10

2. State the test you are about to conduct (this will come in hand when we learn various intervals and inference tests)0 Ex) One proportion z-interval

3. Show your formula and plug-in for your z-interval

4. Report your findings. Write a sentence explaining what you found.0 EX) “We are 95% confident that the true proportion of men who

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

Confidence Interval Example0 An experiment finds that 27% of 53 subjects report improvement after

using a new medicine. 0 Create a 95% confidence interval for the actual cure rate. Use z* = 1.96.

  

 0 Why is this interval so wide?

 

Confidence Interval Example (cont.)

0 An experiment finds that 27% of 53 subjects report improvement after using a new medicine. 0 Make it narrower – 90% confidence.

 

0 What are the advantages and disadvantages?       

Choosing Your Sample Size

0 The question of how large a sample to take is an important step in planning any study.

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

0 The formula requires which we don’t have yet because we have not taken the sample. A good estimate for , which will yield the largest valuefor (and therefore for n) is 0.50.

0 Solve the formula for n. ME z*p̂q̂

n

p̂q̂

p̂p̂

Sample Size Example0 Recall: the Fox news poll which estimated that 82% of all voters believed

global warming exists had a margin of error of 3%. Suppose an environmental group planning a follow-up survey of voters’ opinions on global warming wants to determining a 95% confidence interval with a margin of error of no more than 2%. How large a sample do they need? Use the Fox News estimate as a basis for your calculation.

Sample Size Example #2A 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?

What Can Go Wrong?

Don’t Misstate What the Interval Means:0 Don’t suggest that the parameter varies.0 Don’t claim that other samples will agree with yours.0 Don’t be certain about the parameter.0 Don’t forget: It’s about the parameter (not the

statistic).0 Don’t claim to know too much.0 Do take responsibility (for the uncertainty).0 Do treat the whole interval equally.

What Can Go Wrong? (cont.)

Choosing Your Sample Size:0 In general, the sample size needed to produce a

confidence interval with a given margin of error at a given confidence level is:

where z* is the critical value for your confidence level.

0 To be safe, round up the sample size you obtain.

n z 2 p̂q̂ME2

Recap

0 Finally we have learned to use a sample to say something about the world at large.

0 This process (statistical inference) is based on our understanding of sampling models, and will be our focus for the rest of the book.

0 In this chapter we learned how to construct a confidence interval for a population proportion.0 Best estimate of the true population proportion is the one

we observed in the sample.

Recap (cont.)

0 Best estimate of the true population proportion is the one we observed in the sample.

0 Create our interval with a margin of error.0 Provides us with a level of confidence.0 Higher level of confidence, wider our interval.0 Larger sample size, narrower our interval.0 Calculate sample size for desired degree of precision

and level of confidence.0 Check assumptions and condition.

Recap (cont.)

0 We’ve learned to interpret a confidence interval by Telling what we believe is true in the entire population from which we took our random sample. Of course, we can’t be certain, but we can be confident.

Assignments: pp. 455 – 458

0Day 1: # 1, 3a, 3b, 5, 7, 9, 11, 23

0Day 2: # 4a, 10, 12, 13, 15, 19, 22, 27, 29

0Day 3: # 6, 8, 14, 18, 32