Question

Independent simple random samples of four regions gave the data on race as shown in the table. At the 1 % significance​ level, do the data provide sufficient evidence to conclude that a difference exists in race distributions among the four​ regions?

	white	black	other	total
Northeast	99	11	5	115
Midwest	119	16	9	144
South	162	37	8	207
West	110	13	15	138
Total	490	77	37	604

What are the null and alternative hypotheses?

A.

H0​: The racial distribution is the same in each of the four regions.

Ha​: The racial distribution is not the same in each of the four regions.

B.

H0​: The racial distribution is not the same in each of the four regions.

Ha​: The racial distribution is the same in each of the four regions.

C.

H0​: The racial distribution is the same in at least two of the four regions.

Ha​: The racial distribution is not the same in at least two of the four regions.

D.

H0​: The racial distribution is not the same in at least two of the four regions.

Ha​: The racial distribution is the same in at least two of the four regions.

Test statistic is: ________

Find the critical value: ______

What can we conclude?

Answer 1

The null and alternative hypothesis are,

A)

H₀: The racial distribution is the same in each of the four regions.

H₁: The racial distribution is not the same in each of the four regions.

The test statistic is,

Where, O_i= Observed Frequency

E_i= Expected Frequency

The observed and expected frequencies are given by,

	White	Black	Other	Total
Northeast	99	11	5	115
Midwest	119	16	9	141
South	162	37	8	207
West	110	13	15	138
Total	490	77	37	604

The expected frequency is given by,

	White	Black	Other
Northeast	93.29	14.66	7.05
Midwest	114.39	17.97	8.64
South	167.93	26.39	12.68
West	111.95	17.59	8.45

Then the test statistic is,

Degrees of freedom=(row-1)*(coloumn-1)=(4-1)*(3-1)=3*2= 6

From table with and 6 degrees of freedom,

Critical value,

We reject H₀ if

Here, 14.7873<16.812. That is,

Hence we fail to reject H₀.

Then we can conclude that,

The racial distribution is same in each of the four regions.

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