Question

The age distribution of the Canadian population and the age distribution of a random sample of 455 residents in the Indian community of a village are shown below.

Age (years)	Percent of Canadian Population	Observed Number in the Village
Under 5	7.2%	47
5 to 14	13.6%	78
15 to 64	67.1%	282
65 and older	12.1%	48

Use a 5% level of significance to test the claim that the age distribution of the general Canadian population fits the age distribution of the residents of Red Lake Village.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: The distributions are the same.
H₁: The distributions are different.

H₀: The distributions are different.
H₁: The distributions are the same.    

H₀: The distributions are different.
H₁: The distributions are different.

H₀: The distributions are the same.
H₁: The distributions are the same.

(b) Find the value of the chi-square statistic for the sample. (Round your answer to three decimal places.)


Are all the expected frequencies greater than 5?

Yes No    

What sampling distribution will you use?

chi-square

binomial    

uniform

normal

Student's t

What are the degrees of freedom?


(c) Estimate the P-value of the sample test statistic.

P-value > 0.100 0.050 < P-value < 0.100    

0.025 < P-value < 0.050

0.010 < P-value < 0.025

0.005 < P-value < 0.010

P-value < 0.005

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis that the population fits the specified distribution of categories?

Since the P-value > α, we fail to reject the null hypothesis.

Since the P-value > α, we reject the null hypothesis.   

Since the P-value ≤ α, we reject the null hypothesis.

Since the P-value ≤ α, we fail to reject the null hypothesis.

(e) Interpret your conclusion in the context of the application.

At the 5% level of significance, the evidence is insufficient to conclude that the village population does not fit the general Canadian population.

At the 5% level of significance, the evidence is sufficient to conclude that the village population does not fit the general Canadian population.    

Answer 1

(a)

(i)

The level of significance = = 0.05

(ii)

Correct option:

H₀: The distributions are the same.
H₁: The distributions are different.

(b)

(i)

The value of the chi-square statistic for the sample is got as follows:

Age	Observed (O)	Expected (E)	(O - E)2/E
under 5	47	455 X 7.2/100 = 32.76	6.1898
5 to 14	78	455 X 13.6/100 = 61.88	4.1993
15 to 64	282	455 X 67.1/100 = 305.305	1.7790
65 and older	48	455 X 12.1 /100 = 55.055	0.9041
Total = =			13.072

So,

The value of the chi-square statistic for the sample = 13.072

(ii)

All the expected frequencies greater than 5

Correct option:

Yes

(iii)

sampling distribution will you use

Correct option:

chi-square

(iv)

the degrees of freedom = n - 1 = 4 - 1 = 3

So,

the degrees of freedom = 3

(c)
By Technology, p - value = 0.0045

So,

Correct option:

P-value < 0.005

(d)

Correct option:

Since the P-value ≤ α, we reject the null hypothesis.

(e)

Correct option:

At the 5% level of significance, the evidence is sufficient to conclude that the village population does not fit the general Canadian population.