Answer to Question #200890 in Statistics and Probability for Aiyat Ghulam

Question

A research team studying the relationship between blood type and severity of a certain condition in a population collected data on 1500 subjects as displayed in the contingency table shown as Table 1. Table 1 Severity Of Condition Blood Type A B AB O Absent 543 211 90 476 Mild 44 22 8 31 Severe 28 9 7 31 The researcher wished to know if these data were compatible with the hypothesis that severity of condition and blood type are independent. 

Answer 1

$H_0:$  the severity of condition and blood type are independent

$H_a:$ a the severity of condition and blood type are dependent

Compute the expected frequency for each cell.

The expected frequencies are obtained by multiplying the total number of observations by each cell probability:

$E_i= np_i$

Thus the expected number of persons who have blood type A and feel Absent of severity conditions in our sample:

$1500\cdot\frac{615}{1500}\cdot\frac{1320}{1500}=541.2$

the expected frequency of any cell :

$Expected \ frequency=\frac{(Row\ Total)(Column\ Total)}{Grand\ Total}$

Thee the test statistic:

$\chi^2=\sum \frac{(O_i-E_i)^2}{E_i}$

where O_i is observed frequencies,

E_i is expected frequencies.

degrees of freedom:

$df=(r-1)(c-1)=(3-1)(4-1)=6$

the critical value for $\alpha=0.05$ from the table:

$\chi^2_{0.05}=12.59$

Since the value of the test statistic is less than the critical value, we fail to reject the null hypothesis at the 0.05 level of significance. 

We conclude that that severity of the condition and blood type are independent, or simply there is no relationship between the severity of the condition and blood type, at least for these data.