10.1 Power Point

8/7/2019 10.1 Power Point

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Confidence Intervals: The Basics

Section 10.1


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How long can you expect a AAA battery tolast? What proportion of college

undergraduates have engaged in bingedrinking? Is caffeine dependence real?

These are the types of questions that we would

like to be able to answer, but it just isntpractical to ask/experiment on every battery,college undergrad, or caffeine addictedperson.

Instead we select a sample and collect datafrom those individuals only. The goal is toinfer from the sample data some conclusion

about the population.


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We cannot ever be certain that our

conclusions are correct since a different

sample would generally lead to a differentconclusion.

Statistical inference uses the language of

probability to express the strength of ourconclusions. Probability allows us to take

chance variation into account and to correct

our judgment using calculations.


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In the Next Two Chapters

We will study the two most common methods

of statistical inference, confidence intervals

and significance tests.

Both are based on the sampling distributions

of statistics.

That is, both report probabilities that state

what would happen if we used the inference

method many, many times.


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Your Data is Only as Good as Your

Collection Methods

The methods of formal inference require the long-run, regular behavior that probability describes.

Inference is most reliable when the data are

produced by a properly randomized design. If thisisnt done, your conclusions will be open tochallenge.

Formal inference cannot remedy basic flaws in

producing data, such as voluntary responsesamples and uncontrolled experiments. Use thecommon sense you acquired over the first ninechapters and proceed to formal inference onlywhen you are satisfied that the data deserve such

analysis.


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Lets Pretend

To begin learning about the methods of inference, wewill pretend we know the true population standarddeviation , although we would never actually knowthat value without knowing . Once we know thebasic ideas, we will get rid of this unrealisticrequirement.

There are libraries full of more elaborate statisticaltechniques than we will use, but informed use of anyof these methods requires an understanding of theunderlying reasoning.

A computer or calculator can do the arithmetic, butwe must still exercise judgment based on

understanding.


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Hey Baby, Whats Your IQ?

Big City University would like to find the

average IQ of its 5000 freshman. Having

each one take an IQ test would be difficult

and expensive though, so the university

decides to administer the test to an SRS of

50 freshman. The university finds that themean IQ score for this sample is

Lets ponder a few questions

112.x !


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Find Your Exact Match! Really?

Is the mean IQ score of all Big CityUniversity freshman exactly 112?

Probably not. But the law of large numbers tellsus that the sample mean from a large SRS willbe close to the unknown population parameter.Because , we guess that issomewhere around 112.

How close to 112 is likely to be?

Well, to answer this we must ask anotherquestion

112x !


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We ask How would the sample mean vary if we

took many, many samples of 50 freshman from this

same population?

The sampling distribution of describes how the values

of vary in repeated samples. Remember from last

chapter:

The mean of the sampling distribution of is the same as theunknown mean of the entire population.

The standard deviation of for an SRS of 50 freshman is ,

where is the standard deviation of the IQ scores from all Big

City University freshman. Suppose we know that the IQ scores

have standard deviation = 15, then the standard deviation ofis

The central limit theorem tells us that the mean of 50 scores has

a distribution that is close to Normal.

x

x

x

x

x 50

W

15 2.1.50

}


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Put It All Together and What Do

You Have?

Putting these facts together gives us the

reasoning of statistical estimation in a nutshell for

this example:

1. To estimate the unknown population mean ,

use the mean of the SRS

2. Although is an unbiased estimator of, it will

rarely be exactly equal to , so our estimate hassome error.

3. In repeated samples, the values of follow an

approximately Normal distribution with mean

and standard deviation 2.1.

x

x

.x


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4. The 68-95-99.7% Rule says that in about 95% of all

samples, the mean IQ score for the sample will be

within 4.2 (two standard deviations) of the population

mean .

5. Whenever is within 4.2 points of, is within 4.2

points of . This happens in about 95% of all

samples. So the unknown population parameterlies between and in about 95% of all

samples.

6. So if we estimate that , lies somewhere in theinterval 112 4.2 = 107.8 to 112 + 4.2 = 116.2, we

would be calculating this interval using a method that

captures the true in about 95% of all possible

samples.

x

4.2x

x

x

4.2x


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The Big Picture - The big idea is that thesampling distribution of tells us how big the error

is likely to be when we use as an estimate for.

x

x


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What Was That Again?

We have just learned that in 95% of all

samples of 50 Big City University freshman,

the interval will contain the truepopulation mean .

The language of statistical inference usesthis fact about what would happen in many

samples to express our confidence in the

results of any one sample.

4.2x s


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So To Finally Answer the

Question

Earlier we asked How close to 112 is likely

to be? The resulting interval is 112 4.2,

which can be written as (107.8, 116.2). Wecan now say that We are 95% confident that

the unknown mean IQ score for all Big City

University freshman is between 107.8 and

116.2.

Remember this phrasing because we will use

it every time we create a confidence interval.


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Be Careful!

Be sure that you understand the basis for our

confidence. There are only two possibilities:

1. The interval (107.8, 116.2) contains the true .

2. Our SRS was one of the few samples for which

is not within 4.2 points of the true . (Only 5% of all

samples give such inaccurate results.)

We cannot know whether our sample is one of the

unlucky 5%.The phrase we are 95% confident is

shorthand for saying, We got these numbers by a

method that gives correct results 95% of the time.

x


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Twenty-five

samples fromthe same

population give

these 95%

confidenceintervals. In the

long run, 95% of

all samples give

an interval thatcontains the

population mean

.

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