Section 15.1Inference for Population Spread

AP Statistics

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, , mean

, , standard deviation

ˆ , , proportion

, , coefficent of correlation

, , vertical intercept of LSRL

, , slope of LSRL

x

s

p p

r

a

b

3

Why are changes in standard deviation important? Manufacturer’s need to know that if

changes in their processes cause less variation in their outcomes.

For example, even though two processes produce the same mean amount of ice cream, the better process has smaller variation.

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Warning!!!

Procedures for inference about the population mean are extremely sensitive to non-normal distribution

That is to say, the presence of a non-normal distribution make your conclusion wildly inaccurate even in the presence of large samples

That is to say, proceed with extreme caution

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The F test for comparing two standard deviations

1 1 2 2

1 2

Suppose we start with two distinct populations

, and ,

And we don't know either means or standard deviations.

We pull two SRSs from the populations of size and .

As we do the analysis of th

N N

n n

0 1 2

1 2

e variation, we make the typical

null hypothesis claim of "There is no difference."

:

The typical alternate claim is...

:A

H

H

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The F statistic

2122

1 2

If independent SRSs are taken

from two normal distributions

distribution is distribution of the s calculated

from every possible sample size of and .

The distribution short hand is ,

sF

s

F F

n n

F F j

1

2

where

1 degrees of freedom and

1 degrees of freedom.

k

j n

k n

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Comments about the F statistic If sample distributions have

the same standard deviation, the value of F is 1.

In order to make the table of values smaller and easier to read, we only calculate the F value with the larger standard deviation on top.

2122

sF

s

8

Comments about the F statistic Because the variances are

always positive, F is always positive.

The F distribution is skew right.

2122

sF

s

9

Example

An SRS of size 10 is drawn from normal distribution. Its standard deviation is 6.

An SRS of size 11 is drawn from normal distribution. Its standard deviation is 5.

2

2

2

2

5 25.694

6 36

6 361.44

5 25

F

F

10

What is evidence of different standard deviations? When the standard deviations are very

different, the F value will be much larger than 1.

The F table gives the upper tail probabilities that two samples with the same standard deviation would give such different results by pure chance.

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F(9,10)

2

2

2

2

5 2510,9 , .694,

6 36

6 36(9,10), 1.44

5 25.694 1.44

F F

F F

P F P F

Degree of freedom for the

numerator

Degree of freedom for the

denominator

12

1 1 2 2

2

2

4, 14, 16, 20

1619,13 15,12 , 16

416 is below .001

s n s n

F F F

P F

13

1 1 2 2

2

2

6, 14, 4, 20

613,19 10,15 , 2.25

42.25 is between .010 and .050

s n s n

F F F

P F

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Example

Medical experiment to compare the mean effects of calcium and a placebo on the blood pressure of black men. We might also compare the standard deviations to see whether calcium changes the spread of blood pressures among black men.

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Example

Group 1 (Calcium) results: 7, -4, 18, 17, -3, -5, 1, 10, 11, -2n=10, x-bar=5.000, s=8.743

Group 2 (Placebo) results:-1, 12, -1, -3, 3, -5, 5, 2, -11, -1, -3n=11, x-bar=-0.273, s=5.901

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1

2

0 1 2

1 2

Population : Black men taking calcium

Population : Black men taking placebo

Parameter of interest: standard deviation

of the difference in blood pressure

:

:A

H

H

1 1 2 2

2

2

8.743, 10, 5.901, 11

8.7432.195

5.9019,10 ,

2* 2.195 is greater than 2*.1=.2

s n s n

F

F

P F

0

Because the p-value is so large, we fail to

reject the H . There is insufficient evidence

to suggest that calcium effects the standard

deviation of the blood pressure.

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Assignment

15.1 to 15.7 odd