ARE YOU HERE?

1. Yes, and I’m ready to learn

2. Yes, and I need a nap

3. No

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APPLE PROBLEM When a truck load of apples arrives at a packing

plant, a random sample of 125 is selected and examined for bruises, discoloration, and other defects.

The whole truckload will be rejected if more than 5% of the sample is unsatisfactory.

Suppose that in fact 9% of the apples on the truck do not meet the desired standard.

What is the probability that the shipment will be accepted anyway.

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STANDARD DEVIATION

ˆ pqSD p

n

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

For means

SD yn

WHAT IS THE PROBABILITY THAT THE SHIPMENT WILL BE ACCEPTED ANYWAY?

1. 0.0622. 1-0.0623. 04. 15. -1.54

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STANDARD DEVIATION VS. STANDARD ERROR

We don’t know p, μ, or σ, we’re stuck, right?

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

Sample statistics are notated as: s,

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

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STANDARD ERROR

ˆ ˆˆ

pqSE p

n

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For a sample proportion, the standard error is

For the sample mean, the standard error is

sSE y

n

EXCITING STATISTICS ABOUT ISU STUDENTS -2011 DATA

69.1% of sexually active students use condoms

American College Health Association

n=272

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A CONFIDENCE INTERVAL

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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

A CONFIDENCE INTERVAL

p̂

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p̂

p̂

CERTAINTY VS. PRECISION

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CERTAINTY VS. PRECISION

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).

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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.

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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 horizontal line), while others do not:

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SALES PROBLEM

A catalog sales company promises to deliver orders placed on the Internet within 3 days.

Follow-up calls to a few randomly selected customers show that a 95% CI for the proportion of all orders that arrive on time is 81% ± 4%

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WHICH OF THE FOLLOWING STATEMENTS IS CORRECT?

1. Between 77% and 85% of all orders arrive on time.

2. One can be 95% confident that the true population percentage of orders place on the Internet that arrive within 3 days is between 77% and 85%

3. One can be 95% confident that all random samples of customers will show that 81% of orders arrive on time

4. 95% of all random samples of customers will show that between 77% and 85% of orders arrive on time. Slide

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ONE-PROPORTION Z-INTERVAL

ˆ ˆp z SE p

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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.

ˆ ˆ( )ˆ pqSE p n

Z* IS THE CRITICAL VALUE

80% z*=1.282 90% z*=1.645 95% z*=1.96 98%z*=2.326 99% z*=2.576

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CRITICAL VALUES (CONT.)

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

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BIASED SURVEY PROBLEM

Often, on surveys there are two ways of asking the same question.

1) Do you believe the death penalty is fair or unfairly applied?

2) Do you believe the death penalty is unfair or fairly applied?

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BIASED SURVEY PROBLEM

Survey

1) n=597 2) n=597

For the second phrasing, 45% said the death penalty is fairly applied.

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SUPPOSE 54% OF THE RESPONDENTS IN SURVEY #1 SAID THE DEATH PENALTY WAS FAIRLY APPLIED. DOES THIS FALL WITHIN A 95% CONFIDENCE INTERVAL FOR SURVEY #2?

1. Yes, it falls within my CI2. No, it does not fall within my CI

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MARGIN OF ERROR: CERTAINTY VS. PRECISION

The more confident we want to be, the larger our z* has to be

But to be more precise (i.e. have a smaller ME and interval), we need a larger sample size, n.

We can claim, with 95% confidence, that the interval contains the true population proportion. 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.Slide 1- 22

2 ( )ˆ ˆp SE p

p̂

MARGIN OF ERROR - PROBLEM

Suppose the truth is that 56% of ISU student drink every weekend.

We want to create a 95% confidence interval, but we also want to be as precise as possible.

How many people should we sample to get a ME of 1%?

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HOW MANY PEOPLE SHOULD WE SAMPLE TO GET A ME OF 1%?

1. 1,0002. Between 1,000 and 4,0003. Between 4,000 and 8,0004. Between 8,000 and 16,000

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UPCOMING WORK

Quiz #4 in class today

HW #8 due Sunday

Part 3 of Data Project due Oct. 28th