Question

A multinomial experiment produced the following results: (You may find it useful to reference the appropriate table: chi-square table or F table)

Category	1	2	3	4	5
Frequency	87	52	89	88	79

a. Choose the appropriate alternative hypothesis to test if the population proportions differ.

a All population proportions differ from 0.20.

b Not all population proportions are equal to 0.20.

b. Calculate the value of the test statistic. (Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)

c. Find the p-value.

a 0.01  p-value < 0.025

b p-value < 0.01

c p-value  0.10

d 0.05  p-value < 0.10

e 0.025  p-value < 0.05

d. Can we conclude at the 5% significance level that the population proportions are not equal?

a No, since the p-value is less than the significance level.

b Yes, since the p-value is less than the significance level.

c No, since the p-value is more than the significance level.

d Yes, since the p-value is more than the significance level.

Answer 1

Solution:

Part a. Choose the appropriate alternative hypothesis to test if the population proportions differ.

We have k = 5 categories , thus proportion for each category = 1 / 5 = 0.20

Thus alternative hypothesis is:

b Not all population proportions are equal to 0.20.

( Note under alternative hypothesis, we never say all proportions are different from stated proportion, we say at least one of the proportion is different from stated proportion)

Part b. Calculate the value of the test statistic.

Chi square test statistic for goodness of fit

Where

O_i = Observed Counts

E_i =Expected Counts

Thus we need to make following table:

Category	Oi: Observed frequency	Ei: Expected Frequency	Oi2/Ei
1	87	79	95.810
2	52	79	34.228
3	89	79	100.266
4	88	79	98.025
5	79	79	79.000
	N = 395

Thus

Part c. Find the p-value.

df = k- 1 = 5 - 1 = 4

Look in Chi-square table for df = 2 row and find an interval in which Chi-square test statistic = fall,
then find corresponding right tail area interval ( in ascending order) , which is p-value interval.

Chi-square test statistic = fall between 11.143 and 13.277

corresponding right tail area interval ( in ascending order) is between 0.010 and 0.025

thus p-value interval is:

0.01 < p-value < 0.025

Thus correct answer is:

a 0.01  p-value < 0.025

Part d. Can we conclude at the 5% significance level that the population proportions are not equal?

Since 0.01  p-value < 0.025 < 0.05 significance level, we reject the null hypothesis.

Thus we conclude that: the population proportions are not equal.

Thus correct answer is:

b Yes, since the p-value is less than the significance level.