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8.4 Expected Counts in Two Way Tables

2 min readโ€ขjanuary 7, 2023

Josh Argo

Josh Argo

Jed Quiaoit

Jed Quiaoit


AP Statisticsย ๐Ÿ“Š

265ย resources
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Tests from Two-Way Tables

Another form of data that we can use ฯ‡2 tests for involves data from a two-way table (like the one pictured below). ๐ŸŽจ
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Image Taken from MathBitsNotebook

When performing a ฯ‡2 test from a two way table, there are two different tests we may have to perform and choosing which one can be tricky.

Test for Homogeneity

A chi-squared test for homogeneity is used to compare the distribution of a categorical variable between two or more independent groups or populations. It is used to test the null hypothesis that the proportions of the categories are the same in all the groups. In other words, we'll have to use the chi-squared test for homogeneity when we're comparing two different populations and if two different populations have different amounts for a given categorical variable. ๐Ÿž

Test for Independence

A chi-squared test for independence is used to examine the relationship between two categorical variables in a single population. It is used to test the null hypothesis that the two variables are independent, meaning that the presence or absence of one variable does not affect the probability of the other variable. In other words, we'll use this test when we're comparing within one population to see if two categorical variables are associated within the one population, we would use a ฯ‡2 test for independence. ๐Ÿ’ฅ

Expected Counts

Regardless of which test we are doing, we will be comparing two multi-row/column matrices rather than just two rows or columns. This means we have to calculate the expected counts matrix based off of our observed counts table. This will be done by doing the following for each cell:ย ๐Ÿ˜€
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-nAEnbgtrwxab.png?alt=media&token=20b4173d-6af1-4b9a-bb69-9eafbb4bc603
where n = table total.

Example

In the first cell on the two way table above on SUV and sports car ownership in regard to male or female, we would take the total male (60) and the total SUV (156), multiply those to get 9360. Then we would divide that total by our table total (240) and get 39. We would complete this process for all steps to create our expected count table. ๐Ÿš—
Our final expected counts answer would be:
SUV
Sports Car
Male
39
21
Female
117
63

Applications

In the next sections, we will talk about setting up the two types of ฯ‡2 tests regarding a two way table and how we will use these observed and expected counts to determine if we have association or homogeneity between our two variables/populations. Stay tuned 'til then! ๐Ÿ˜‰
๐ŸŽฅย  Watch: AP Stats Unit 8 - Chi Squared Tests

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๐Ÿ‘†Unit 1 โ€“ Exploring One-Variable Data
โœŒ๏ธUnit 2 โ€“ Exploring Two-Variable Data
๐Ÿ”ŽUnit 3 โ€“ Collecting Data
๐ŸŽฒUnit 4 โ€“ Probability, Random Variables, & Probability Distributions
๐Ÿ“ŠUnit 5 โ€“ Sampling Distributions
โš–๏ธUnit 6 โ€“ Proportions
๐Ÿ˜ผUnit 7 โ€“ Means
โœณ๏ธUnit 8 โ€“ Chi-Squares
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