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### Calculating and Interpreting Expected and Chi-Square Tables

```
Date: 04/10/98 at 22:31:30
From: Lola Kelly
Subject: Stats- chi squared

How do you get the frequency expected when doing chi squared? I have
```

```
Date: 04/11/98 at 08:06:14
From: Doctor Anthony
Subject: Re: Stats- chi squared

I will work through a 3x3 contingency table. The method will be
directly applicable to any similar problem. The following table refers
to the performance of a baseball team on various pitches classified as
good, medium and bad. The null hypothesis is that the state of the
pitch does not affect the performance of the team.

Wins     Draws     Losses     Total
---------|--------------------------|----------
Good    | 11        6          4   |     21
Medium   | 12        7          7   |     26
Bad    |  7        7         14   |     28
---------|--------------------------|-----------
Total    | 30       20         25   |     75

To find the expected frequencies, we assume independence of the rows
and columns. To get the expected frequency corresponding to the 11 at
top left, we look at row total (21) and column total (30), multiply
them, and then divide by the overall total (75). So the expected
frequency is:

21*30
-------  = 8.4
75

So to complete the expected table, draw up another table similar to
that above and having the same row and column totals. For each entry
in this table, we simply calculate (row total*column total)/75. The
completed table is:

Wins     Draws     Losses     Total
-----------|--------------------------|-----------
Good    | 8.4      5.6        7.0  |    21
Medium   |10.4      6.9        8.7  |    26
Bad    |11.2      7.5        9.3  |    28
-----------|--------------------------|------------
Total    | 30       20         25   |    75

The number of degrees of freedom is calculated for an m-by-n table as
(m-1)(n-1), so in this case (3-1)(3-1) = 2*2 = 4.

To calculate X^2, we then have a further table:

O       E       |O-E|       |O-E|^2/E
----------------------------------------
11      8.4       2.6         0.805
6      5.6       0.4         0.0286
4       7         3          1.2857
12     10.4       1.6         0.246
7      6.9       0.1         0.0014
7      8.7       1.7         0.332
7     11.2       4.2         1.575
7      7.5       0.5         0.033
14      9.3       4.7         2.375
-------------------------------------------
Total = 6.70  = X^2

The tabular 95% value of X^2 (degrees of freedom = 4) is 9.49, so the
value of X^2 that we obtained (6.70) is not significant at the 5%
level. We conclude that the state of the pitch does not affect the
performance of the team.

-Doctor Anthony, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```

```
Date: 04/11/98 at 13:00:10
From: Coolbabe29
Subject: Re: Stats- chi squared

```
Associated Topics:
High School Statistics

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