Question

Suppose you conducted a drug trial on a group of animals and you hypothesized that the animals receiving the drug would show increased heart rates compared to those that did not receive the drug. You conduct the study and collect the following data:
Calculate ${\chi }^2$ statistics.

Answer 1

Let us find the row and column totals and total frequency from the given $2\times 2$ contingency table.

Expected frequencies are given by

$E_{ij}=\frac{R_i\times C_j}{N}$

Therefore,

$E_{11}=\frac{R_1\times C_1}{N}=\frac{50\times 66}{105}=31.42$,

$E_{12}=\frac{R_1\times C_2}{N}=\frac{50\times 39}{105}=18.57$

$E_{21}=\frac{R_2\times C_1}{N}=\frac{55\times 66}{105}=34.57$,

$E_{22}=\frac{R_2\times C_2}{N}=\frac{55\times 39}{105}=20.42$

Write the table of expected frequencies

We have,

${\chi }^2=\sum \left[\frac{(O_{ij}-E_{ij}{)}^2}{E_{ij}}\right]$

$=\frac{(36-31.42{)}^2}{31.42}+\frac{(14-18.57{)}^2}{18.57}+\frac{(30-34.57{)}^2}{34.57}+\frac{(25-20.42{)}^2}{20.42}$

$=\frac{20.97}{31.42}+\frac{20.88}{18.57}+\frac{20.88}{34.57}+\frac{20.97}{20.42}$

$=0.66+1.12+0.60+1.02$

${\chi }^2=3.4$.