Overview of the Chi-Square DistributionChapter 10: Chi-Square And F Distributions

Part 1: Inferences Using the Chi-Square Distribution

Overview of the Chi-Square Distribution

• The Chi-Square distribution is a probability distribution.• Other probability distributions we’ve been working with are the

normal distribution and the Student’s t distribution.

• Chi is a Greek Letter denoted • Chi-Square is denoted

Overview of the Chi-Square Distribution

• The distribution is of values• It starts at 0 and contains positive values• It is NOT symmetric• It is dependent on degrees of freedom

• As the degrees of freedom increase, the graph of the chi-square distribution become bell-like and begins to look more and more symmetric

• The mode (high point) of the chi-square distribution with degrees of freedom occurs over (for )

Overview of the Chi-Square Distribution

• Table C in the AP Stats Formula Sheet shows critical values of chi-square distributions for which a designated area falls to the right of the critical value

Degrees of Freedom

Area in the right tail

Critical Values

10.1 Chi-Square: Tests of Independence and of HomogeneityChapter 10: Chi-Square And F Distributions

Part 1: Inferences Using the Chi-Square Distribution

Chi-Square: Tests of Independence• We are testing the independence of two factors

• a chi-square test for independence is applied when you have two categorical variables from a single population. It is used to determine whether there is a significant association between the two variables.

• Hypotheses• The variable A and variable B are independent• The variable A and variable B are not independent

Chi-Square: Tests of Independence• A contingency table is used to record the observed frequencies and

expected frequencies when comparing two factors. • The boxes that contain the observed frequencies are called cells• The size of a contingency table is rows columns

• The observed frequency is the number of actual observed data points that share the two factors being compared.

• When we are testing for independence, we use the null hypothesis to determine the expected frequency of each cell.• The expected frequency

• Note: If the expected value is not a whole number, do NOT round it to the nearest whole number

Keyboard versus Time to Learn to Type at 20 wpm

Table 10-2

Chi-Square: Tests of Independence• The sample test statistic is • The value is a measure of the sum of the differences

between observed frequency and expected frequency in each cell.

• The smaller the value of the test statistic, the higher the agreement between the expected and observed counts. Larger values of the test statistic indicate more discrepancy between the observed and expected counts.

Degrees of Freedom

• number of cell rows• number of cell columns

Chi-Square: Tests of Independence

• For tests of independence, we always use a right-tailed test on the chi-square distribution because we are testing to see if the measure of the difference between the observed and expected frequencies is too large to be due to chance alone.

Chi-Square: Tests of Independence

How to Test for Independence of Two Statistical Variables1. State the hypotheses and identify the level of significance

• The variable A and variable B are independent• The variable A and variable B are not independent

2. Construct a contingency table (the rows represent one statistical variable and the columns represent the other). Compute the expected frequency for each cell.

• Note: the sample size must be “sufficiently large” meaning that for each cell,

3. Compute the sample test statistic(using the Ti-83/84)

4. Find the P-Value (using the Ti-83/84)

5. Conclude the Test• Compare the P-value to

6. Interpret your results