MATB344 Applied StatisticsMATB344 Applied Statistics

I. Experimental Designs for Small SamplesI. Experimental Designs for Small SamplesII. Statistical Tests of SignificanceII. Statistical Tests of SignificanceIII. Small Sample Test StatisticsIII. Small Sample Test Statistics

Chapter 10Inference from Small Samples

IntroductionIntroduction• When the sample size is small, the

estimation and testing procedures of Chapter 8 and 9 are not appropriate.

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

population means2, the variance of a normal populationThe ratio of two population variances.

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.n

xz/

normal!not is

/ nsx

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 Student’s t distribution, distribution, with nn-1 degrees of freedom.-1 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 of freedom, degrees 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.

t.025 = 2.262

Column subscript = a = .025

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

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/

2/

ndftt

nstx

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.

specified) as working(not Hspecified) as (working H

a

0

15:15:

ExampleExampleData:Data: 17, 31, 12, 17, 13, 25First, calculate the sample mean and standard deviation, using your calculator or the formulas in Chapter 2.

387.75

61152477

1

)(

167.196

115

222

nnxx

s

nxx i

ExampleExampleData:Data: 17, 31, 12, 17, 13, 25Calculate the test statistic and find the rejection region for =.05.

516138.16/387.715167.19

/0

ndf

nsxt

:freedom of Degrees :statisticTest

Rejection Region: Reject H0 if t > 2.015. If the test statistic falls in the rejection region, its p-value will be less than a = .05.

ConclusionConclusionData:Data: 17, 31, 12, 17, 13, 25Compare the observed test statistic to the rejection region, and draw conclusions.

15:15:

a

0

H H

Conclusion: For our example, t = 1.38 does not fall in the rejection region and H0 is not rejected. There is insufficient evidence to indicate that the average activation time is greater than 15.

015.2

38.1

t

t

if HReject :Region Rejection

:statisticTest

0

Approximating the Approximating the pp-value-value• You can only approximate the p-value

for the test using Table 4.

Since the observed value of t = 1.38 is smaller than t.10 = 1.476,

p-value > .10.

The exact The exact pp-value-value• You can get the exact p-value using some calculators or a computer.

One-Sample T: TimesTest of mu = 15 vs mu > 15

Variable N Mean StDev SE MeanTimes 6 19.17 7.39 3.02

Variable 95.0% Lower Bound T PTimes 13.09 1.38 0.113

p-value = .113 which is greater than .10 as we approximated using Table 4.

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

1. Single random sample: The sampled population must be normal.

II. Statistical Tests of SignificanceII. Statistical Tests of Significance1. Based on the t, distributions2. Use the same procedure as in Chapter 93. Rejection region—critical values and significance levels:based on the t, distributions with the appropriate degrees of freedom4. Tests of population parameters: a single mean, the difference between two means, a single variance, and the ratio of two variances

III. Small Sample Test StatisticsIII. Small Sample Test StatisticsTo test one of the population parameters when the sample sizes are small, use the following test statistics: