copyright ©2011 nelson education limited inference from small samples chapter 10

Copyright ©2011 Nelson Education Limited

Inference from Small Samples

CHAPTER 10CHAPTER 10


IntroductionIntroduction• When the sample size is small, the estimation and

testing procedures for sample averages of Chapters 8/9 are not appropriate.

• There are equivalent small sample test and estimation procedures for , the mean of a normal populationthe difference between two population

means


The Sampling DistributionThe Sampling Distribution of the Sample Mean of the Sample Mean

• When we take a sample from a normal population, the sample mean has a normal distribution for any sample size n, and

• has a standard normal distribution. • But if is unknown, and we must use s to estimate it, the

resulting statistic is not normalis not normal.

nxz

/

normal!not is / ns

x

€

X


Student’s t DistributionStudent’s t Distribution• Fortunately, this statistic does have a sampling

distribution that is well known to statisticians, called the Student’s t distribution, Student’s t distribution, with nn-1 -1 degrees of freedom.degrees of freedom.

nsxt/

•We can use this distribution to create estimation testing procedures for the population mean .


Properties of Student’s Properties of Student’s tt

• Shape depends on the sample size n or the degrees degrees of freedom, of freedom, nn-1.-1.

• As n increases the shapes of the t and z distributions become almost identical.

•Mound-shapedMound-shaped and symmetric about 0.•More variable than More variable than zz, with “heavier tails”


Using the Using the tt-Table-Table• Table 4 gives the values of t that cut off certain

critical values in the tail of the t distribution.• Index dfdf and the appropriate tail area a a to find ttaa,,the

value of t with area a to its right.

For a random sample of size n = 10, find a value of t that cuts off .025 in the right tail.

Row = df = n –1 = 9

t.025 = 2.262

Column subscript = a = .025


Small Sample Inference Small Sample Inference for a Population Mean for a Population Mean • The basic procedures are the same as those used for

large samples. For a test of hypothesis:

.1on with distributi- taon basedregion rejection aor values- using

/

statistic test theusing tailedor two one :H versus:HTest

0

a00

ndfp

nsxt


• For a 100(1)% confidence interval for the population mean

.1on with distributi- ta of tailin the /2 area off cuts that of value theis where 2/

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Small Sample Inference Small Sample Inference for a Population Mean for a Population Mean


ExampleExampleA sprinkler system is designed so that the average time for the sprinklers to activate after being turned on is no more than 15 seconds. A test of 5 systems gave the following times:

17, 31, 12, 17, 13, 25 Is the system working as specified? Test using = .05. Also, calculate a 95\% confidence interval.


Testing the Difference Testing the Difference between Two Meansbetween Two Means

normal. bemust spopulation two thesmall, are sizes sample theSince

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and21

variancesand 2

and 1

means with 2 and 1 spopulation from

drawn are 2

and 1

size of samples randomt independen 9,Chapter in As

μμ

nn

•To test: •H0:D0 versus Ha:one of three

where D0 is some hypothesized difference, usually 0.


Testing the Difference Testing the Difference between Two Meansbetween Two Means•The test statistic used in Chapter 9

•does not have either a z or a t distribution, and cannot be used for small-sample inference. •We need to make one more assumption, that the population variances, although unknown, are equal.

2

22

1

21

21z

ns

ns

xx


Testing the Difference Testing the Difference between Two Meansbetween Two Means•Instead of estimating each population variance separately, we estimate the common variance with

2)1()1(

21

222

2112

nn

snsns

21

2

021

11

nns

Dxxt has a t distribution with n1+n2-2 degrees of freedom.

•And the resulting test statistic,


Estimating the Difference Estimating the Difference between Two Meansbetween Two Means

•You can also create a 100(1-)% confidence interval for 1-2.

2)1()1( with

21

222

2112

nn

snsns

21

22/21

11)( nn

stxx

Remember the three assumptions:

1. Original populations normal

2. Samples random and independent

3. Equal population variances.


ExampleExample• Two training procedures are compared by

measuring the time that it takes trainees to assemble a device. A different group of trainees are taught using each method. Is there a difference in the two methods? Use = .01.

Time to Assemble

Method 1 Method 2

Sample size 10 12Sample mean 35 31Sample Std Dev

4.9 4.5


Testing the Difference Testing the Difference between Two Meansbetween Two Means•How can you tell if the equal variance assumption is reasonable?

statistic. test ealternativan use

,3smaller larger ratio, theIf

.reasonable is assumption varianceequal the

,3smaller larger ratio, theIf

:Thumb of Rule

2

2

2

2

ss

ss


Testing the Difference Testing the Difference between Two Meansbetween Two Means•If the population variances cannot be assumed equal, the test statistic

•has an approximate t distribution with degrees of freedom given above. This is most easily done by computer.

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22

1

21

21

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1)/(

1)/(

2

22

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1

21

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2

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df


The Paired-DifferenceThe Paired-Difference Test Test

•Sometimes the assumption of independent samples is intentionally violated, resulting in a matched-pairsmatched-pairs or paired-difference testpaired-difference test.•By designing the experiment in this way, we can eliminate unwanted variability in the experiment by analyzing only the differences,

ddii = = xx11ii – – xx22ii

•to see if there is a difference in the two population means,


ExampleExample

• One Type A and one Type B tire are randomly assigned to each of the rear wheels of five cars. Compare the average tire wear for types A and B using a test of hypothesis.

Car 1 2 3 4 5Type A 10.6 9.8 12.3 9.7 8.8

Type B 10.2 9.4 11.8 9.1 8.3

• But the samples are not independent. The pairs of responses are linked because measurements are taken on the same car.


The Paired-DifferenceThe Paired-Difference Test Test

.1on with distributi- taon basedregion rejection aor value- theUse

. s,difference theofdeviation standard andmean theare and pairs, ofnumber where

/0

statistic test theusing 0 :H test we0:H test To d0210

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dsdn

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i

d

d


Some NotesSome Notes

•You can construct a 100(1-)% confidence interval for a paired experiment using

•Once you have designed the experiment by pairing, you MUST analyze it as a paired experiment. If the experiment is not designed as a paired experiment in advance, do not use this procedure.

nstd d

2/


ExampleExample•An experimenter has performed a lab experiment using two groups of rats. He wants to test H0:

Standard (2) Experimental (1)

Sample size 10 11Sample mean 13.64 12.42Sample Std Dev 2.3 5.8


Small samples?Small samples?• We have developed methods when sample sizes

are small. To do this, we assume that the underlying distribution is normal.

• What if we’re wrong? Fortunately, the t-test is fairly ROBUST against moderate deviations from normality.


Key ConceptsKey ConceptsI. Experimental Designs for Small SamplesI. Experimental Designs for Small Samples

1. Single random sample: The sampled population must be normal.2. Two independent random samples: Both sampled populations must be normal.

a. Populations have a common variance 2.b. Populations have different variances

3. Paired-difference or matched-pairs design: The samples are not independent.


Key Concepts Key Concepts (ignore last two lines)(ignore last two lines)

copyright ©2011 nelson education limited inference from small samples chapter 10

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