swm02_19.ppt

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Chapter 19

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A Confidence Interval

 Recall that 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:

ˆ

p

pq n

ˆ ˆ

( )

ˆ

pq

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Slide 19- 4

A Confidence Interval (cont.)

 By the 68-95-99.7% Rule, we know

 about 68% of all samples will have ’s within 1

SE of p

 about 95% of all samples will have ’s within 2

SEs of p

 about 99.7% of all samples will have ’s within

3 SEs of p

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

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A Confidence Interval (cont.)

 Consider the 95% level:

 There’s a 95% chance that p is no more than 2

SEs away from .

 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.

This is called a 95% confidence interval.

ˆ

p

ˆ

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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

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

 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

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Critical Values (cont.)

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

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Assumptions and Conditions

 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?

 Success/Failure Condition: We must expect at least 10

“successes” and at least 10 “failures.”

 Plausible Independence Condition: Is there any reason

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Example 1: Local Newspaper

The Houston Chronicle polls a random sample of 330 voters, finding 144 who say they will vote “yes” on the upcoming school budget. Create a 95 % confidence interval for

actual sentiment of all voters.

Plausible Independence Condition Random Sampling Condition

10% Condition

Success Failure Condition

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Activity – Coin Flipping

Let’s work together to create and graph confidence intervals for each of your samples.

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What Does “95% Confidence” Really Mean?

(cont.)

 The figure to the

right shows that some of our

confidence

intervals capture the true proportion (the green

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Example 2: New Medicine

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.

 Make it wider – 90% confidence interval.

 What are the advantages and the

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Optional – Hershey’s Kisses

Toss (gently) a Hershey’s kiss 60

times. Record the number of times that it lands base down.

Calculate your sample proportion for landing base down.

Create a 90% confidence interval for landing base down. (Make sure that you check conditions!!!)

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 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.

 

2

ˆ ˆ

z

pq

n

ME





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Example 3: More Medicine

Remember the example about the experiment with a certain medicine? (Example 2) What sample size would we need in a follow-up study if we want a margin of error of 5% with 98%