stats exam 2 - chi-squared

Chi-Squared Distribution

When there is a small DF, the shape of the x² distribution is

positively skewed

As DF increases, the shape of the x² distribution becomes more

symmetrical

The mean of a x² distribution is equal to its

df

E(X²)=

df

var(X²)=

2df

std(X²)=

√2df

Three ways to use X² tests

1. test for proportions

2. test for independence

3. test for sample variances

Multinomial probabilites and X² distributions are inverses of each other - how?

Multinomial probabilities determine how likely your data is, given population info and X² determines how likely are the stated population info, given your data.

When would the X² statistic you’ve calculated NOT follow the X² distribution?

When the expectation in any cell, m_i, becomes too small (too small means less than 5 or 1)

As N gets larger…

…X² statistic is better approximated by a X² distribution

If your expectation in any cell (m_i) gets too small, how do you fix this?

Increase N!

A small X² is indicative of…

…good fit

What does it mean when X² is 0?

The observed data and expectation are a complete match

How do you find the df for X²?

k-1, where k is the number of categories

Why is df for X² k-1?

We can determine all the other expected counts from one, thereforce one set of expected counts canNOT vary

What is x² test of variance achieving?

Assessing if variance in a single population is different from a prespecified or hypothesized value - connected to z-scores!!

What is x² statistic?

The sum of independent, squared z-scores

In x² tests of variance, why is df equal to n-1?

Because the last score isn’t free to vary as the deviations from the mean must equal 1

x² is proportional to the ratio of

sample variance to hypothesized population variance

When the null hypothesis is true for a x² test of variance, the s²/σ² ratio should be..

…near 1

For a one-tailed x² test of variance, which page of the X² table do we use?

Second page

For a two-tailed x² test of variance, which page of the X² table do we use?

First and second page!

How can you confirm if you have the right critical value for your test?

If H_A states <, then the critical value < df.
If H_A states >, then the critical value > df.