Chi-square Goodness-of-fit Test

Goodness-of-fit Test is good for testing a One-row Frequency Table.
The test shows how well certain proportions fit our sample, which only has ONE variable(row).

Steps:

• Choose a distribution (depends on the DF)
• Complete the Frequency Table with Expected Counts for each data
• Calculate the standardized Chi-square value 𝐗² according to the Expected & Observed
• Use calculator to get the probability area P-value in the distribution according to 𝐗²

Expected Frequencies

Counting the Expected Frequency for data is the very first step and fundamental part of doing Chi-square Test.

The expected frequencies can be either a PRESET or the PROBABILITY of the data.
The expected frequencies are set as Null Hypothesis in the test,
and Observed frequencies are the Alternative Hypothesis against the null in the test.

“For a χ² goodness-of-fit test, the null hypothesis is that the population distribution of the categorical variable in question matches some hypothesized distribution. We use that hypothesized distribution to calculate the expected counts for each value of the variable.”

Example

Solve:

• Since the null hypothesis is 4 feeders has “equally likely” chance to feed the bird,
• so the supposed chance for each feeder would be 350/4 = 87.5
• and 87.5 is the Expected count for each feeder.

Test statistic 𝐗²

Chi-squared Test statistic Formula:

To calculate 𝐗², we need to COMPLETE the _Frequency Table_, with both Expected and Observed values:

Or you can see it as:

P-value

To calculate P-value we need the 𝐗² and _DF_:

For instance, it observes 3 prices for a fruit: prices of apple, orange, banana. Then there are 3 categories, or 3 variables. Therefore the DF (Degree of freedom) is (3-1)=2

Get an online chi-squared calculator, input the test statistic 𝐗² and DF, we'll get its P-value, like this:

Example

Solve:

• Calculate the expected frequencies for each data:

• Compare Observed and expected, to get Chi-squared:

• Get a P-value calculator, input test statistic 𝐗²=10.5 and Degree of freedom df= 4-1 = 3:

Making conclusions in a goodness-of-fit Test

To overturn the __null hypothesis__, we just to compare the P-value with Significance Level.

But there’s another type of conclusion we can make: which component contributes the most to the test statistic.
The way to do it, is simply look at each component’s value, the bigger component the more it contributes.

Example

Solve:
District B has the largest component because its observed count was farthest away from its expected count (relative to the expected count). So we can say that District B contributed the most to the 𝐗² test-statistic.