Question

You may need to use the appropriate technology to answer this question.

Test the following hypotheses by using the

χ²

goodness of fit test.

H0:	pA = 0.40, pB = 0.40, and pC = 0.20
Ha:	The population proportions are not pA = 0.40, pB = 0.40, and pC = 0.20.

A sample of size 200 yielded 60 in category A, 120 in category B, and 20 in category C. Use α = 0.01 and test to see whether the proportions are as stated in

H₀.

(a)

Use the p-value approach.

Find the value of the test statistic.

__________

Find the p-value. (Round your answer to four decimal places.)

p-value =

Repeat the test using the critical value approach.

Find the value of the test statistic.

____________

State the critical values for the rejection rule. (If the test is one-tailed, enter NONE for the unused tail. Round your answers to three decimal places.)

test statistic≤:

test statistic≥:

Answer 1

Solution:

Given:

H0:	pA = 0.40, pB = 0.40, and pC = 0.20
Ha:	The population proportions are not pA = 0.40, pB = 0.40, and pC = 0.20.

A sample of size 200 yielded 60 in category A, 120 in category B, and 20 in category C.

Level of significance =  α = 0.01

Part a) Use the p-value approach.

Find the value of the test statistic.

Chi square test statistic for goodness of fit

Where

O_i = Observed Counts

E_i =Expected Counts = N * Expected proportions = 200 * Expected proportions

Thus we need to make following table

Category	Oi: Observed Frequencies	Expected Proportions	Ei: Expected Frequencies	Oi2/Ei
A	60	0.40	80	45.000
B	120	0.40	80	180.000
C	20	0.20	40	10.000
	N = 200

Thus

Find the p-value.

df = k - 1 = 3 - 1 = 2

Use excel command:

=CHiSQ.DIST.RT( x , df )

=CHiSQ.DIST.RT( 35.000 , 2 )

=0.0000

Thus p-value = 0.0000

Repeat the test using the critical value approach.

Find the value of the test statistic.

State the critical values for the rejection rule.

Use excel command:

=CHISQ.INV.RT( probability, df)

=CHISQ.INV.RT(0.01,2)

=9.210

Thus Critical value = 9.210

test statistic ≤ : NONE

test statistic ≥ : 9.210

Decision:

Since test statistic value = 35.000 > critical value = 9.210, we reject null hypothesis H0.

Conclusion:

At 0.01 level of significance , we have enough evidence to reject null hypothesis that: p_A = 0.40, p_B = 0.40, and p_C = 0.20