BRM - Lesson7 Confidence Interval

8/10/2019 BRM - Lesson7 Confidence Interval

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Lesson 7:Confidence intervals


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Introduction

The difference between the present and the new commercial may be aresult of chance.

The example involves inferences about a population based on sampledata How can we analyse such problems?

Two equivalent methods:1. Confidence Intervals: estimating a value of a population parameter

2. Tests of significance: assessing evidence for a claim about a

population

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Remember the sample distribution of The sample mean is a random variable (if you pick a different sample

you will probably get a different sample mean).

The distribution of:

has mean and standard deviation

- If the population distribution is normal, then the distribution of is

normal.

- If the population distribution is not normal and n is large (in practice:

0), then is approximately normal (the Central Limit Theorem).

We assume that the population is known.

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Statistical confidence Example 6.5:

In a sample of 453 large firms the mean monthly premium paid

for the health insurance is 0 dollars. We assume that is

known to be 112.

What can we say about the population mean ?

In repeated sampling is approximately

,

,

11

, .

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Statistical confidence is approximately, .

The 69-95-99.7 rule says that the probability is

about 0.95 that is within 10.54 of .

2 3 + + 2 + 3

The 68-95-99.7 rule

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Statistical confidence To say that lies within 10.54 of is the same as saying that is

within 10.54 of ! (because distances are symmetrical)

So in 95% of all samples is within 10.54 of or put differently: in 95%

of all samples is in the interval from 10. to 10.

(with 0).

We say that we are 95% confident

that is between 394 and 416.

We call 0 10. , 1

a 95% confidence interval

for.

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Statistical confidence We are 95% confident that is between 394 and 416.

Be sure you understand that in reality could be outside this interval!

(Although this only happens in 5% of all samples)

We cannot know whether our sample is one of the 95% or one of the

unlucky 5%. The only thing we can say is: we arrived at the conclusion

that is between 394 and 416 by a process that gives correct results

95% of the time.

Implication: We dont need to take a lot of random samples to find the

sampling distribution and at its center. All we need is one sample and

relying on the properties of the sample means distribution to infer the

population mean .

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Confidence levels and critical values A confidence level C (e.g. 95%) gives the probability that the

interval will capture the true parameter in repeated samples.

The critical value z* for a confidencelevel C is the value so that the area

under the standard Normal curve

between z* and z* is C.

Common critical values:

o z*=2.576 (for C=99%)

o z*=1.960 (for C=95%)

o z*=1.645 (for C=90%)

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Confidence levels and critical values Exercise: Find the critical value for a 70% confidence level.

End solution:

z*=1.04

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Confidence intervals A confidence interval always has the form

where the margin of errorm shows the precision of the estimate.

The confidence interval for a population mean with a known andwith a confidence level Cis:

where

the margin of error.

To derive the confidence interval we had to assume that the distributionofis normal. So the interval is exact when the population distribution

is normal and is approximately correct when n is large in other cases

(because of the central limit theorem).

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Confidence intervalsA confidence interval is an

interval that contains the true

parameter in C percent of the

samples.

Experiment with the applet:http://bcs.whfreeman.com/psbe3e/#613741__630698__

Figure 6.5: Twenty-five samples from thesame population gave these 95%

confidence intervals.

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Application: quality controlA company makes 00 bottles of green tea. Assume we knowthe standard deviation is . In a sample of 10 bottles, themean content 01.. Is this convincing evidence that the

mean fill of all bottles differs from 00?

Two common mistakes: do not conclude that

- the mean is 01. therefore it is proven that the mean differs

from 00.- the difference of 1. is small compared to 00 so there isnothing unusual going on.

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Application: quality controlSo: 01. 10

Is this convincing evidence that the mean fill differs from 00?

We calculate a 95% confidence interval for :

C.I. for

01. 1.

00. , 0.

Since 500 is not included in the C.I., we are 95% confident that differs

from 500. So we can reject the hypothesis that the mean fill of all bottles

is 00.

Inferential analysis can be done either with confidence intervals or

significance tests (next lesson).

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Reducing the margin of errorConfidence interval for:

To reduce the margin of error (to make the confidence interval

smaller), there are only 3 ways:

1. Use a lower level of confidence (smaller C smaller )

2. Reduce

3. Increase the sample size

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Choosing the minimum sample sizeIt is possible to choose the sample size in a research so that inferences

about the population mean can be done with a predetermined level of

confidence and a predetermined margin of error:

Note: in practice is usually unknown, but if n is large then the sample standard

deviation s will be close to the unknown population standard deviation . Sosubstituting s for in the formula will result in an approximately correct minimum

sample size n.

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Exercise 1In a sample of 50 Belgian employees, the mean

gross monthly wage is 00 euro. The standard

deviation in the population is known to be 1000 euro.

a. Find the 80% confidence interval for.

b. Find the 95% confidence interval for.c. Find the 95% confidence interval for if the sample size

in this example 500 instead of 50.

d. What sample size would you need to limit the margin of

error to 50 euro with 95% confidence?

End solutions:

a. (2219,2581) b. (2123,2677) c. (2312,2488) d. 1536 cases

Confidence interval for:

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Exercise 2 An opinion poll asks:

Will you vote for Barack Obama in the elections: Yes or No?

Let be a variable taking the values 0 (No) and 1 (Yes), so that isthe percentage of Obama voters in the sample.

In a sample of 2400 respondents

o 0. (49% of the respondents in the sample say they will voteObama)

o Assume that standard deviation is 0.0.

Estimate the margin of error in this poll for a 95% confidence level.

End solution:

the margin of error in this poll is approximately 2 percentage points.

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BRM - Lesson7 Confidence Interval

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