8 Student’s t test

7/28/2019 8 Students t test

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Students t test

This test was invented by astatistician working for the

brewer Guinness. He was calledWS Gosset (1867-1937), butpreferred to keep anonymous so

wrote under the nameStudent.


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The t-distribution

William Gossetlived from 1876 to 1937

Gosset invented the t-test to handle small samples for quality

control in brewing. He wrote under the name "Student".


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t-Statistic

ns

xt

/

When the sampled population isnormally distributed, the t statistic isStudent t distributed with n-1 degrees

of freedom.


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T-test1. Test for single mean

Whether the sample mean is equal to the predefinedpopulation mean ?

2. Test for difference in means

Whether the CD4 level of patients taking treatment A is

equal to CD4 level of patients taking treatment B ?

3. Test for paired observationWhether the treatment conferred any significant benefit ?


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T- test for single meanThe following are the weight (mg) of each of 20rats drawn at random from a large stock. Is it

likely that the mean weight for the whole stock

could be 24 mg, a value observed in some previous

work?.

9 18 21 26

14 18 22 27

15 19 22 29

15 19 24 30

16 20 24 32


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Steps for test for single mean1. Questioned to be answered

Is the Mean weight of the sample of 20 rats is 24 mg?N=20, =21.0 mg, sd=5.91 , =24.0 mg

2. Null Hypothesis

The mean weight of rats is 24 mg. That is, Thesample mean is equal to population mean.

3. Test statistics --- t (n-1) df

4. Comparison with theoretical value

if tab t (n-1) < cal t (n-1) reject Ho,

if tab t (n-1) > cal t (n-1) accept Ho,

5. Inference

ns

xt

/

x


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t test for single mean

Test statisticsn=20, =21.0 mg, sd=5.91 ,=24.0 mg

t

= t .05, 19 = 2.093 Accept H0 if t < 2.093

Reject H0 if t >= 2.093

x

30.22091.5

240.21

llt

Inference :

There is no evidence that the sample is taken

from the population with mean weight of 24 gm


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-1.

96

0

Area = .025

Area =.005

Z

-2.

575

Area = .025

Area = .005

1.

96

2.

575

Determining the p-Value


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.9

5

t0

f(t)

-1.96 1.96

.025025

red area = rejection region for 2-sided test


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Given below are the 24 hrs total energyexpenditure (MJ/day) in groups of lean and

obese women. Examine whether the obesewomens mean energy expenditure issignificantly higher ?.

Lean6.1 7.0 7.5

7.5 5.5 7.6

7.9 8.1 8.1

8.1 8.4 10.2

10.9

T-test for difference in means

Obese8.8 9.2 9.2

9.7 9.7 10.0

11.5 11.8 12.8


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Two sample t-test

Difference

between means

Sample size

Variability

of data

t-test t++


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T-test for difference in means

Data Summarylean Obese

N 13 9

8.10 10.30

S 1.38 1.25

82.3

13

25.1

9

32.1

3.101.822

llt

Inference : The cal t (3.82) is higher than tab t at0.05, 20. ie 2.086 . This implies that there is a

evidence that the mean energy expenditure in obese

group is significantly (p


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Example

Suppose we want to test the

effectiveness of a program designedto increase scores on the quantitativesection of the Graduate Record Exam

(GRE). We test the program on agroup of 8 students. Prior to enteringthe program, each student takes a

practice quantitative GRE; aftercompleting the program, each studenttakes another practice exam. Basedon their performance, was the

program effective?


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Each subject contributes 2 scores:repeated measures design

Student Before Program After Program

1 520 555

2 490 510

3 600 585

4 620 645

5 580 630

6 560 550

7 610 645

8 480 520

h d i h i l


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Can represent each student with a singlescore: the difference (D) between thescores

Student

Before Program After Program

D1 520 555 35

2 490 510 20

3 600 585 -15

4 620 645 25

5 580 630 50

6 560 550 -10

7 610 645 35

8 480 520 40


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Approach: test the effectiveness ofprogram by testing significance of D

Alternative hypothesis: program iseffective scores after program willbe higher than scores before

program average D will be greaterthan zero

H0: D 0H1: D > 0

S d t k D d D2


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Student

Before

Program

After

Program D D2

1 520 555 35 1225

2 490 510 20 400

3 600 585 -15 225

4 620 645 25 625

5 580 630 50 2500

6 560 550 -10 100

7 610 645 35 1225

8 480 520 40 1600

D = 180 D2 = 7900

So, need to know D and D2:


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Recall that for single samples:

errorstandard

mean-score

X

obt

s

Xt

For related samples:

D

D

obt

s

Dt

where:

N

ss

D

D and

1

2

2

N

N

DD

sD


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45.23

18

81807900

1

22

2

N

N

DD

sD

5.22

8

180

N

DD

Standard deviation of D:

Mean of D:

Standard error:

2908.88

45.23

N

ss

D

D


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D

D

obt

s

Dt

Under H0, D = 0, so:

714.2

2908.8

5.22

D

obt

s

Dt

From Table B.2: for = 0.05, one-tailed, with df = 7,

tcrit

= 1.895

2.714 > 1.895 reject H0

The program is effective.


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t-Valuet is a measure of:

How difficult is it to believe the null hypothesis?

High t

Difficult to believe the null hypothesis -accept that there is a real difference.

Low tEasy to believe the null hypothesis -

have not proved any difference.


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In Conclusion !

Student s t-test will be used:

--- When Sample size is small

and for the following situations:

(1) to compare the single sample meanwith the population mean

(2) to compare the sample means of

two indpendent samples

(3) to compare the sample means ofpaired samples

8 Student’s t test

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