Question

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

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	64	37	67	59	57

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

• All population proportions differ from 0.20.

• 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.

• 0.025 p-value < 0.05
• 0.01 p-value < 0.025

• p-value < 0.01

• p-value 0.10
• 0.05 p-value < 0.10

d. Can we conclude at the 10% significance level that the population proportions are not equal?

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

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

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

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

Answer 1

Answer: A multinomial experiment produced the following results:

Solution:

Proportion p = 1/5 = 0.20

a) The null and alternative hypothesis test:

Ho: p1 = p2 = p3 = p4 = p5 = 0.20

Ha: Not all population proportions are equal to 0.20.

b) From the given data we used chi square test statistic χ2 at α = 0.05 significance level.

n = 64+37+67+59+57

n = 284

Expected frequency = n*p

Expected frequency = 284 * 0.20

Expected frequency = 56.8

Category	Observed (O)	Expected (E)	(O-E)^2/E
1	64	56.8	0.9127
2	37	56.8	6.9021
3	67	56.8	1.8317
4	59	56.8	0.0852
5	57	56.8	0.0007
Total	284	284	9.7324

Test statistic χ2 = Σ(O-E)^2/E

Test statistic χ2 = 9.732

c) P-value:

Degree of freedom, df = k-1

df = 5 - 1 = 4

P-Value is .0452

Since, P-Value is 0.0452.

The first option is correct.

0.025 < P-value (0.0452) < 0.05

d) Conclusion:

Since, P-value (0.0452) < α (0.10)

Reject the null hypothesis Ho. There is sufficient evidence to conclude that the not all population proportions are equal to 0.20.