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 120 in category A, 20 in category B, and 60 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 =

State your conclusion.

Reject H₀. We conclude that the proportions differ from 0.40, 0.40, and 0.20.Reject H₀. We conclude that the proportions are equal to 0.40, 0.40, and 0.20.    Do not reject H₀. We cannot conclude that the proportions are equal to 0.40, 0.40, and 0.20.Do not reject H₀. We cannot conclude that the proportions differ from 0.40, 0.40, and 0.20.

(b)

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≥

State your conclusion.

Do not reject H₀. We cannot conclude that the proportions differ from 0.40, 0.40, and 0.20.Reject H₀. We conclude that the proportions are equal to 0.40, 0.40, and 0.20.    Do not reject H₀. We cannot conclude that the proportions are equal to 0.40, 0.40, and 0.20.Reject H₀. We conclude that the proportions differ from 0.40, 0.40, and 0.20.

Answer 1

Solution:-

a)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: p_A = 0.40, p_B = 0.40, and p_C = 0.20

Alternative hypothesis:The population proportions are not p_A = 0.40, p_B = 0.40, and p_C = 0.20.

Formulate an analysis plan. For this analysis, the significance level is 0.01.

Analyze sample data. Applying the chi-square goodness of fit test to sample data, we compute the degrees of freedom, the expected frequency counts, and the chi-square test statistic. Based on the chi-square statistic and the degrees of freedom, we determine the P-value.

DF = k - 1 = 3 - 1
D.F = 2
(E_i) = n * p_i


X² = 75.0

where DF is the degrees of freedom, k is the number of levels of the categorical variable, n is the number of observations in the sample, E_i is the expected frequency count for level i, O_i is the observed frequency count for level i, and X² is the chi-square test statistic.

The P-value is the probability that a chi-square statistic having 2 degrees of freedom is more extreme than 75.0.

We use the Chi-Square Distribution Calculator to find P(X² > 75.0) = 0.00

Interpret results. Since the P-value (0.00) is less than the significance level (0.01), we cannot accept the null hypothesis.

Reject H₀. We conclude that the proportions differ from 0.40, 0.40, and 0.20.

b)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: p_A = 0.40, p_B = 0.40, and p_C = 0.20

Alternative hypothesis:The population proportions are not p_A = 0.40, p_B = 0.40, and p_C = 0.20.

Formulate an analysis plan. For this analysis, the significance level is 0.01.

Analyze sample data. Applying the chi-square goodness of fit test to sample data, we compute the degrees of freedom, the expected frequency counts, and the chi-square test statistic. Based on the chi-square statistic and the degrees of freedom, we determine the P-value.

DF = k - 1 = 3 - 1
D.F = 2
(E_i) = n * p_i


X² = 75.0

X²_Critical = 9.221

Rejection region is t > 9.221

where DF is the degrees of freedom, k is the number of levels of the categorical variable, n is the number of observations in the sample, E_i is the expected frequency count for level i, O_i is the observed frequency count for level i, and X² is the chi-square test statistic.

Interpret results. Since the X²-value (75.0) lies in the rejection region, hence, we cannot accept the null hypothesis.

Reject H₀. We conclude that the proportions differ from 0.40, 0.40, and 0.20.