Lecture 21

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COMPARING TWO POPULATIONS

IDEA: Compare two groups/populations based on samples from each of them.

Examples.

Compare average height of men and women. Draw sample of men heights: x1, x2, …, xm and a sample of women heights: y1, y2, …, yn.

Test Ho:

Ho:

μx = μy vs Ha:

μx ≠ μy or Ha:

μx > μy

Compare proportions of Democrats in two cities,

Compare weights of people before and after a diet, etc.

General considerations for the samples : Dependent or independent samples.

Example. Comparing weights of people before and after a diet we have dependent (same people ) samples of weights. Comparing weights of people in two cities we have independent samples. Analysis methods will differ for dependent and independent samples.

PAIRED t-TEST: dependent samples

Observations come as matched pairs (X,Y).

X and Y are NOT independent, X and Y are dependent .

Examples.

X is score on a test before studying studying hard for the same student; hard; Y is score on the test after

X is score on a test or in sports before training after training program; program, Y score

X is weight before program; weight loss program, Y is weight after the

X and Y are heights of twins or siblings.

PAIRED t-TEST: HYPOTHESES

Hypotheses of interest : does training make a difference?

μx = score before training; μy = score after training.

Ho:

μx = μy vs Ha:

μx < μy

(no difference) (score after training is higher)

Data are pairs of observations

: (x1, y1), (x2, y2), …, (xn, yn).

Typically, we work with differences: d=X-Y , and phrase hypotheses in terms of differences:

μd

= true mean difference

.

In terms of differences:

Hypotheses e.g. Ho:

μd = 0

vs

Ha:

μd < 0

Data: d1, d2, …, dn.

obs before after difference

1

2

.

n

.

x1 x2 xn

.

y1 y2 yn

.

d1=x1-y1 d2=x2=y2 dn=xn-yn

PAIRED t-TEST: TEST PROCEDURE

To test Ho, we do one sample t-test . Need sample mean and standard deviation of d’s:

n d

1

n i n

1

d i

and

s

2

d

i

1

(

d n i

1

d

)

2

.

Compute the test statistic :

t

s d

/

d n

.

Under Ho the test statistic has t(n-1) distribution .

Make decision in exactly the same way as for the one sample t-test.

A (1-

α)100% CI for d:

d

t

/ 2

(

n

1)

s d n

.

PAIRED t-TEST: an example

The amount of lactic acid in the blood was examined for 10 men, before and after a strenuous exercise, with the results in the following table.

(a) Test if exercise changes the level of lactic acid in blood. Use significance level

α=0.01.

(b) Find a 95% CI for the mean change in the blood lactose level.

Before

After

15

33

16

20

13

30

13

35

17

40

20

37

13

18

16

26

14

21

18

19

PAIRED t-TEST: lactic acid example contd.

Solution

. Take d=“After level” – “before level” of lactic acid.

Data for d: 18, 4, 17, 22, 23, 17, 5, 10, 7, 1. Sample stats:

d

12.4 and

s d

2

63.156.

STEP1.

Ho:

μd = 0 vs Ha: μd ≠ 0

STEP 2.

Test statistic:

t

s d

/

d n

12.4

7.95 / 10

4.93.

STEP 3 . Critical value? df=n-1=9, t

α/2

=t

0.005

=3.69.

STEP 4 . DECISION: t = 4.93 > 3.69 = t

0.005

, so reject Ho.

STEP 5.

Exercise changes lactic acid level.

Example contd.

(b) Find a 95% CI for the mean change in the blood lactose level.

d

t

/ 2

s d n

It is the familiar formula for the 95% CI for the mean, this time mean difference

μd

. Need percentile from the t distribution with n-1 degrees of freedom.

n=10, n-1=9,

α=0.05, so t

α/2

=t

0.025

=2.262, so the 95% CI for

μd is:

7.947

10

(6.716,18.086).

Lactic acid example in MINITAB: data

set lactic-acid.MPJ

Lactic acid example in MINITAB: data

set

lactic-acid.MPJ

Paired T-Test and Confidence Interval

Paired T for before - after

N Mean StDev SE Mean before 10 15.50 2.37 0.75

after 10 27.90 8.17 2.58

Difference 10 -12.40 7.95 2.51

95% CI for mean difference: (-18.08, -6.72)

T-Test of mean difference = 0 (vs not = 0): T-Value = -4.93 P-Value =

0.001

Ho Ha

Histogram of Differences

(with Ho and 95% t-confidence interval for the mean)

3

Conclusion : Reject Ho, lactic acid level changes after exercise.

Note

: CI for “Before –after”

2

1

0

-25 -20

[

_

X

-15 -10

Differences

]

-5

Ho

0

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