Chapter 9.3 (323) A Test of the Mean of a Normal Distribution: Population Variance Unknown Given a random sample of n observations from a normal population

Chapter 9.3 (323)Chapter 9.3 (323)A Test of the Mean of a Normal Distribution: A Test of the Mean of a Normal Distribution:

Population Variance UnknownPopulation Variance UnknownGiven a random sample of n observations from a normal population with mean . Using the sample mean and standard deviation X and s we can use the following test with significance level ,

(i) To test either null hypothesis

against the alternative

the decision rule is

Or equivalently

0000 :: HorH

,10

0 if HReject ntns/

-μXt

nstXX nc / if HReject ,100

01 : H

A Test of the Mean of a Normal Distribution: A Test of the Mean of a Normal Distribution: Population Variance UnknownPopulation Variance Unknown

(continued)(continued)

(ii) To test either null hypothesis



Or equivalently

0000 :: HorH

,10

0 if HReject ntns/

-μXt

nstXX nc / if HReject ,100

01 : H

A Test of the Mean of a Normal Distribution: A Test of the Mean of a Normal Distribution: Population Variance UnknownPopulation Variance Unknown


(iii) To test the null hypothesis



equivalently

where tn-1,/2 is the student t-value for n – 1 degrees of freedom* and upper tail probability** /2. The p-values for these tests are computed in the same way as we did for tests with known variance except that the student t value is substituted*** for the normal Z value. *frígráður **líkur í efri hala ***sett í stað

00 : H

2/,10

02/,10

0 if HReject if HReject nn tns/

-μXtort

ns/

-μXt

nstXornstX nn / if HReject / if HReject 2/,1002/,100

01 : H

Chapter 9.4Chapter 9.4 (327)(327) Tests of the Population ProportionTests of the Population Proportion

(Large Sample Size) (Large Sample Size)We begin by assuming a random sample of n observations from a population that has a proportion whose members possess a particular attribute (tiltekið viðhorf) . If (1 - ) > 9 and the sample proportion is p the following tests have significance level : (i) To test either null hypothesis

against the alternative (valtilgátunni)

the decision rule is (ákvörðunarreglunni)

0000 :: HorH

Zn

pZ

/)1(

if HReject 00

00

01 : H

Tests of the Population ProportionTests of the Population Proportion (Large Sample Size) (Large Sample Size)

(Continued)(Continued)




0000 :: HorH

Zn

pZ

/)1(

if HReject 00

00

01 : H

Tests of the Population ProportionTests of the Population Proportion (Large Sample Size) (Large Sample Size)

(Continued)(Continued)


against the two-sided alternative


For all of these tests the p-value is the smallest significance level at which the null hypothesis can be rejected.

00 : H

2/

00

002/

00

00

/)1( if HReject

/)1( if HReject

Zn

pZorZ

n

pZ

01 : H

Chapter 9.5Chapter 9.5 (330)(330) Tests of Variance of a Normal PopulationTests of Variance of a Normal Population

Given a random sample of n observations from a normally distributed population with variance 2. If we observe the sample variance sx

2, then the following tests have significance level : (i) To test either the null hypothesis



20

20

20

20 :: HorH

2,12

0

2

0

)1( if HReject

nxsn

20

21 : H

Tests of Variance of a Normal PopulationTests of Variance of a Normal Population(continued)(continued)




20

20

20

20 :: HorH

21,12

0

2

0

)1( if HReject

nxsn

20

21 : H

Tests of Variance of a Normal PopulationTests of Variance of a Normal Population(continued)(continued)




Where 2n-1 is a chi-square random variable and P(2

n-1 > 2n-1,) =

.The p-value for these tests is the smallest significance level at which the null hypothesis can be rejected given the sample variance.

20

20 : H

22/1,12

0

22

2/,120

2

0

)1()1( if HReject

nx

nx sn

orsn

20

21 : H

Some Probabilities for the Chi-Some Probabilities for the Chi-Square DistributionSquare Distribution

(Figure 9.5) (Figure 9.5) Section 9.5 page 331Section 9.5 page 331

/21 -

2v,/20

/2

2v,1-/2

f(2v)

Chapter 9.6Chapter 9.6 (334)(334) Tests of the Difference Between Tests of the Difference Between

Population Means: Matched PairsPopulation Means: Matched Pairs

Suppose that we have a random sample of n matched (samstæð) pairs of observations (mælinga/athugana) from distributions with means X and Y . Let D and sd denote the observed sample mean and standard deviation for the n differences Di = (xi – yi) . If the population distribution of the differences is a normal distribution, then the following tests have significance level .(i) To test either null hypothesis



0000 :: DHorDH yxyx

01 : DH yx

,10

0 if HReject n

D

tn/s

-DD

Tests of the Difference Between Tests of the Difference Between Population Means: Matched PairsPopulation Means: Matched Pairs





0000 :: DHorDH yxyx

01 : DH yx

,10

0 if HReject n

D

tn/s

-DD

Tests of the Difference Between Tests of the Difference Between Population Means: Matched PairsPopulation Means: Matched Pairs



against the two-sided alternative


Here tn-1, is the number for which P(tn-1 > tn-1, ) = where the random variable tn-1 follows a Student’s t distribution with (n – 1) degrees of freedom. When we want to test the null hypothesis that the two population means are equal, we set D0 = 0 in the formulas. P-values for all of these tests are interpreted as the smallest significance level at which the null hypothesis can be rejected (hægt er að hafna) given the test statistic.

00 : DH yx

01 : DH yx

2/,10

2/,10

0 if HReject n

D

n

D

tn/s

-DDort

n/s

-DD

Tests of the Difference Between Tests of the Difference Between Population Means: Independent Population Means: Independent

Samples (Known Variances)Samples (Known Variances)

Suppose that we have two independent random samples of nx and ny observations from normal distributions with means X and Y and variances 2

x and 2y . If the observed sample

means are X and Y, then the following tests have significance level .(i) To test either null hypothesis



0000 :: DHorDH yxyx

01 : DH yx

Z

nn

-DYX

y

y

x

x

22

00 if HReject

Tests of the Difference Between Population Tests of the Difference Between Population Means: Independent Samples (Known Means: Independent Samples (Known

Variances)Variances)(continued)(continued)




0000 :: DHorDH yxyx

01 : DH yx

Z

nn

-DYX

y

y

x

x

22

00 if HReject

Tests of the Difference Between Population Tests of the Difference Between Population Means: Independent Samples (Known Means: Independent Samples (Known

Variances)Variances)(continued)(continued)




If the sample sizes are large (n > 100) then a good approximation at significance level can be made if the population variances are replaced by the sample variances. In addition the central limit leads to good approximations even if the populations are not normally distributed. P-values for all these tests are interpreted as the smallest significance level at which the null hypothesis can be rejected given the test statistic.

00 : DH yx

01 : DH yx

2/22

02/22

00 if HReject

Z

nn

-DYXorZ

nn

-DYX

y

y

x

x

y

y

x

x

Tests of the Difference Between Population Tests of the Difference Between Population Means: Population Variances Unknown and Means: Population Variances Unknown and

EqualEqualThese tests assume that we have two independent random samples of nx and ny observations from normally distributed populations with means X and Y and a common variance. The sample variances sx

2 and sy2 are used to compute a pooled

variance estimator

Then using the observed sample means are X and Y, the following tests have significance level :(i) To test either null hypothesis



0000 :: DHorDH yxyx

01 : DH yx

,222

00 if HReject

yx nn

y

p

x

p

t

n

s

n

s

-DYX

)2(

)1()1( 222

yx

yyxxp nn

snsns


EqualEqual(continued)(continued)




0000 :: DHorDH yxyx

01 : DH yx

,222

00 if HReject

yx nn

y

p

x

p

t

n

s

n

s

-DYX


EqualEqual(continued)(continued)




Here tnx+ny-2, is the number for which P(tnx+ny-2, > tnx+ny-2, ) = .

P-values for all these tests are interpreted as the smallest significance level at which the null hypothesis can be rejected given the test statistic.

00 : DH yx

01 : DH yx

2/,222

02/,222

00 if HReject

yxyx nn

y

p

x

p

nn

y

p

x

p

t

n

s

n

s

-DYXort

n

s

n

s

-DYX


Not EqualNot EqualThese tests assume that we have two independent random samples of nx and ny observations from normal populations with means X and Y and a common variance. The sample variances sx

2 and sy2 are used. The degrees of freedom, v, for the student t

statistic is given by

Then using the observed sample means are X and Y, the following tests have significance level :(i) To test either null hypothesis



0000 :: DHorDH yxyx

01 : DH yx

,22

00 if HReject v

y

y

x

x

t

n

s

ns

-DYX

)1/()()1/()(

)()(

22

22

222

yy

yx

x

x

y

y

x

x

nn

sn

n

s

n

s

n

s

v


Not EqualNot Equal(continued)(continued)




0000 :: DHorDH yxyx

01 : DH yx

,22

00 if HReject v

y

y

x

x

t

n

s

ns

-DYX


Not EqualNot Equal(continued)(continued)




Here tnx+ny-2, is the number for which P(tnx+ny-2, > tnx+ny-2, ) = .

P-values for all these tests are interpreted as the smallest significance level at which the null hypothesis can be rejected given the test statistic.

00 : DH yx

01 : DH yx

2/,22

02/,22

00 if HReject v

y

y

x

x

v

y

y

x

x

t

n

s

ns

-DYXort

n

s

ns

-DYX

Chapter 9.7Chapter 9.7 (346)(346) Testing the Equality of Population Testing the Equality of Population

Proportions (Large Samples)Proportions (Large Samples)

Given independent random samples of nx and ny with proportion successes px and py. When we assume that the population proportions are equal, an estimate of the common proportion is

For large sample sizes - - n(1 - ) > 9 - - the following tests have significance level :(i) To test either null hypothesis



0:0: 00 yxyx HorH

0:1 yxH

Z

npp

npp

pp

yx

yx

)1()1(

)( if HReject

0000

0

yx

yyxx

nn

pnpnp

0

Testing the Equality of Population Testing the Equality of Population Proportions - Large Samples -Proportions - Large Samples -





0:0: 00 yxyx HorH

0:1 yxH

Z

npp

npp

pp

yx

yx

)1()1(

)( if HReject

0000

0

Testing the Equality of Population Testing the Equality of Population Proportions - Large Samples -Proportions - Large Samples -





It is also possible to compute and interpret the p-values for these tests by calculating the minimum significance level at which the null hypothesis can be rejected.

0:0 yxH

0:1 yxH

2/

0000

2/

0000

0)1()1(

)(

)1()1(

)( if HReject Z

npp

npp

pporZ

npp

npp

pp

yx

yx

yx

yx

Chapter 9.8 (350)Chapter 9.8 (350)The F DistributionThe F Distribution

Given that we have two independent random samples of nx and ny observations from two normal populations with variances 2

x and 2y . If the sample variances are sx

2 and sy2

then the random variable

Has an F distribution with numerator degrees of freedom (nx – 1) and denominator degrees of freedom (ny – 1). An F distribution with numerator degrees of freedom v1 and denominator degrees of freedom v2 will be denoted Fv1, v2 . We denote Fv1, v2, the number for which

We need to emphasize that this test is quite sensitive to the assumption of normality.

22

22

/

/

yy

xx

s

sF

)( ,,, 2121 vvvv FFP

Tests for Equality of Variances from Tests for Equality of Variances from Two Normal PopulationsTwo Normal Populations

Let sx2 and sy

2 be observed sample variances from independent random samples of size nx and ny from normally distributed populations with variances 2

x and 2y . Use s2

x to denote the larger variance. Then the following tests have significance level :(i) To test either null hypothesis


the decision rule is ,1,12

2

0 if HReject yx nn

y

x Fs

sF

220

220 :: yxyx HorH

221 : yxH

Tests for Equality of Variances from Tests for Equality of Variances from Two Normal PopulationsTwo Normal Populations


(ii) To test the null hypothesis



Where s2x is the larger of the two sample variances. Since

either sample variance could be larger this rule is actually based on a two-tailed test and hence we use /2 as the upper tail probability. Here Fnx-1,ny-1 is the number for which

Where Fnx-1,ny-1 has an F distribution with (nx – 1) numerator degrees of freedom and (ny – 1) denominator degrees of freedom.

2/,1,12

2

0 if HReject yx nn

y

x Fs

sF

220 : yxH

221 : yxH

)( ,1,11,1 yxyx nnnn FFP

Chapter 9.9Chapter 9.9 (354)(354) Determining the Probability of a Type II Determining the Probability of a Type II

ErrorErrorConsider the test


Using a decision rule

Using the decision rule determine the values of the sample mean that result in accepting the null hypothesis. Now for any value of the population mean defined by the alternative hypothesis H1 find the probability that the sample mean will be in the acceptance region for the null hypothesis. This is the probability of a Type II error. Thus we consider = * such that * > 0. Then for * the probability of a Type II error is

and Power = 1 -

00 : H

01 : H

cXnσ/ZμXZnσ/

-μX or if HReject 02/

00

]/

[)|(*

*

n

XZPXXP c

c

Power Function for Test HPower Function for Test H00: : = 5 = 5 against Hagainst H11: : > 5 ( > 5 ( = 0.05, = 0.05, =0.1, n =0.1, n

= 16)= 16)(Figure 9.13)(Figure 9.13)

5.105.055.000

.05

.5

1

Pow

er (

1 - )

Key WordsKey Words Alternative Hypothesis Determining the

Probability of Type II Error

Equality of Population Proportions

F Distribution Hypothesis Testing

Methodology Interpretation of the

Probability value or p-value

Null Hypothesis Power Function States of Nature and

Decisions on Null Hypothesis

Test of Mean of a Normal Distribution (Variance Known) Composite Null and

Alternative Composite or Simple Null

and Alternative Hypothesis

Key WordsKey Words(continued)(continued)

Testing the Equality of Two Population Proportions (Large Samples)

Tests for Difference Between Population Means: Independent Samples

Tests for Equality of Variances from Two Normal Populations

Tests for the Difference Between Sample Means: Population Variances Unknown and Equal

Tests for Differences Between Population Means: Matched Pairs

Tests of the Mean of a Normal Distribution: Population Variance Unknown

Tests of the Population Proportion (Large Sample Sizes)

Tests of Variance of a Normal Population

Type I Error Type II Error

Documents

Chapter 9.3 (323) A Test of the Mean of a Normal Distribution: Population Variance Unknown Given a random sample of n observations from a normal population