The Z statistic

€

Zx =x − μ x

σ x

€

σX

=σ

n

Where

€

σX

The Z statistic

€

Zx =x − μ x

σ x

€

σX

=σ

n

Where

€

σX

The Z statistic

€

Zx =x − μ x

σ x

€

σX

=σ

n

Where

€

σX

The Z statistic

€

Zx =x − μ x

σ x

€

σX

=σ

n

Where

€

σX

The t Statistic(s)

• Using an estimated , which we’ll call we can create an estimate of which we’ll call

€

ˆ σ X

=ˆ σ

n€

σ 2

€

ˆ σ 2

€

σX

€

ˆ σ X

where

€

ˆ σ =(X i − X )2

n −1∑ =

nS2

n −1

The t Statistic(s)

• Using, instead of we get a statistic that isn’t from a normal (Z) distribution - it is from a family of distributions called t

€

tn−1 =x − μ x

ˆ σ x

€

ˆ σ X

€

σX

The t Statistic(s)

• What’s the difference between t and Z?

The t Statistic(s)

• What’s the difference between t and Z?

• Nothing if n is really large (approaching infinity)– because n-1 and n are almost the same

number!

The t Statistic(s)

• With small values of n, the shape of the t distribution depends on the degrees of freedom (n-1)

The t Statistic(s)

• With small values of n, the shape of the t distribution depends on the degrees of freedom (n-1)– specifically it is flatter but still symmetric

with small n

The t Statistic(s)

• Since the shape of the t distribution depends on the d.f., the fraction of t scores falling within any given range also depends on d. f.

The t Statistic(s)

• The Z table isn’t useful (unless n is huge) instead we use a t-table which gives tcrit for different degrees of freedom (and usually both one- and two-tailed tests)

The t Statistic(s)

• There is a t table on page 142 of your book

• Look it over - notice how tcrit changes with the d.f. and the alpha level

The t Statistic(s)

• The logic of using this table to test alternative hypothesis against null hypothesis is precisely as with Z scores - in fact, the values in the bottom row are given by the Z table and the familiar +/- 1.96 appears for alpha = .05 (two-tailed)

Next Time:

• More about t tests