Question

The following table is from the publication of Vital and Health Statistics from the Centers for Disease Control and Prevention/National Center for Health Statistics. The individuals in the following table have an eye irritation, a nose irritation, or a throat irritation. They have only one of the three. Is there sufficient evidence to reject the hypothesis that the type of ear, nose, or throat irritation is independent of the age group at a level of significance equal to .05?Age (years)

Type of Irritation

18-29 30-44 45-64 65 and Older

Eye 460 567 358 60

Nose 904 1311 793 109

Throat 236 322 159 17

Find Test Statistic

Find P-value 2

Answer 1

Here chi square test of independence will be used

Hypotheses are:

H0: The type of ear, nose, or throat irritation is independent of the age group.

Ha: The type of ear, nose, or throat irritation is not independent of the age group.

Following table shows the row total and column total:

	18-29	30-44	45-64	65 and older	Total
Eye	460	567	358	60	1445
Nose	904	1311	793	109	3117
Throat	236	322	159	17	734
Total	1600	2200	1310	186	5296

Expected frequencies will be calculated as follows:

$E_{ij}=\frac{ith\,row\,total\cdot jth\,row\,total}{grand\,total}$

Following table shows the expected frequency:

	18-29	30-44	45-64	65 and older	Total
Eye	436.556	600.264	357.43	50.75	1445
Nose	941.692	1294.826	771.01	109.472	3117
Throat	221.752	304.909	181.56	25.779	734
Total	1600	2200	1310	186	5296

Following table shows the calculations for test statistics:

O	E	(O-E)^2/E
460	436.556	1.25899343
904	941.692	1.508653428
236	221.752	0.915461885
567	600.264	1.843345088
1311	1294.826	0.202033537
322	304.909	0.957998226
358	357.43	0.000908989
793	771.01	0.627177469
159	181.56	2.80322538
60	50.75	1.685960591
109	109.472	0.002035077
17	25.779	2.989675356
Total		14.79546846

The test statistics

$\chi^{2}=\sum \left ( \frac{(O-E)^{2}}{E} \right )=14.795$

Degree of freedom: df =( number of rows -1)*(number of columns-1) = (3-1)*(4-1)=6

The p-value is:

p-value = 0.0219

Since p-value is less than 0.05 so we reject the null hypothesis.

There is sufficient evidence to reject the hypothesis that the type of ear, nose, or throat irritation is independent of the age group at a level of significance equal to 0.05.