T-distribution & comparison of means Z as test statistic Use a Z-statistic only if you know the...

T-distribution & comparison of meansT-distribution & comparison of means

Z as test statisticUse a Z-statistic only if you know the population standard deviation (σ). Z-statistic converts a sample mean into a z-score from the null distribution.

n

X

SE

XZ HH

test

00

p-value is the probability of getting a Ztest as extreme as yours under the null distribution


t-test: uses sample data to evaluate a hypothesis about a population mean when population stdev () is unknown

We use the sample stdev (s) to estimate the standard error

t as a test statistic

x = n sx = s

n

X

HXz

0

X

H

s

Xt 0

standard error

estimatedstandard

error


You can use s to approximate σ, but then the sampling distribution is a t distribution instead of a normal distributionWhy are Z-scores normally distributed, but t-scores are not?

t distribution

Random variable

Random variable

constant

X

Htest s

Xt 0

X

Htest

Xz

0

Random variable

constant

constant

(normal) (normal)

(non-normal)


t distribution

Very large sample - estimated standard error almost = the true standard error (t almost exactly the same as Z)

t distribution is a family of curves. As n gets bigger, t becomes more normal.

For smaller n, the t distribution is platykurtic (narrower peak, fatter tails)

Use “degrees of freedom” to decide which t curve to use. Basic t-test, df = n-1


t distribution

df .05 .025 .01 .005

1234

6.3142.9202.3532.132

12.7064.3033.1822.776

32.8216.9654.5413.747

63.6579.9255.8414.604

tcrit

Level of significance for a one-tailed test


Practice with Table B.3

With a sample of size 6, what are the degrees of freedom? For a one-tailed test, what is the critical value of t for an alpha of .05? And for an alpha of .01?

Df = 5, tcrit = 2.015; tcrit = 3.365


Practice with Table B.3

For a sample of size 25, doing a two-tailed test, what are the degrees of freedom and the critical value of t for an alpha of .05 and for an alpha of .01?

Df = 24, tcrit = 2.064; tcrit = 2.797


Practice with Table B.3You have a sample of size 13 and you are doing a one-tailed test. Your tcalc = 2. What do you approximate the p-value to be?

p-value between .025 and .05

What if you had the same data, but were doing a two-tailed test?

p-value between .05 and .10


Two-sample testing

All the inferential statistics we have considered involve using one sample as the basis for drawing conclusion about one population.

Most research studies aim to compare of two (or more) sets of data in order to make inferences about the differences between two (or more) populations.

What do we do when our research question concerns a mean difference between two sets of data?


Two-sample testing

Steps for Calculating a Test Statistic:

To test the null hypothesis – compute a t statistic and look up in Table B.3

Ho: 1 - 2 = 0

HA: 1 - 2 0


Two-sample testing


General t formula

t = sample statistic - hypothesized population parameter estimated standard error


Two-sample testing


One Sample t

X

Htest s

Xt 0

Remember?

21

)()( 2121

XXs

XXt

For two-sample test


Two-sample testingSteps for Calculating a Test Statistic:

Standard Error for a Difference in Means21 XXs

The one-sample standard error ( sx ) measures how much error expected between X and .

The two sample standard error (sx1-x2) measures how much error is expected when you are using a sample mean difference (X1 – X2) to represent a population mean difference.


Two-sample testing


2

22

1

21

21 n

s

n

ss XX

Standard Error for a Difference in Means


Two-sample testing


Calculate the total amount of error involved in using two sample means - find the error from each sample separately and then add the two errors together.

Each of the two sample means - there is some error.


Two-sample testing


2

22

1

21

21 n

s

n

ss XX

Standard Error for a Difference in Means

BUT: Only works when n1 = n2


Two-sample testing

Steps for Calculating a Test Statistic:Change the formula slightly - use the pooled sample variance instead of the individual sample variances.

Pooled variance = weighted estimate of the variance derived from the two samples.

sp2

SS1 SS2

df1 df2


One-Sample T

1. Calculate sample mean

2. Calculate standard error

3. Calculate T and d.f.

4. Use Table B.3


Two-Sample T

1. Calculate X1-X2

sp2

SS1 SS2

df1 df2

t (X 1 X 2) (1 2 )

sx 1 x 2 d.f. = (n1 - 1) + (n2 - 1)

sp2

n1

sp

2

n2

2. Calculate pooled variance

3. Calculate standard error

4. Calculate T and d.f.

5. Use Table B.3


EXAMPLE


SUMMARY OF EQUATIONS

T-distribution & comparison of means Z as test statistic Use a Z-statistic only if you know the...

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