Question

In: Statistics and Probability

The age distribution of the Canadian population and the age distribution of a random sample of...

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%                   48            
5 to 14 13.6%                   68            
15 to 64 67.1%                   294            
65 and older 12.1%                   45            

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 different.    

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

H0: The distributions are different.
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?

Student's t

uniform    

binomial

normal

chi-square


What are the degrees of freedom?


(c) Find or estimate the P-value of the sample test statistic. (Round your answer to four decimal places.)


(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.

Solutions

Expert Solution

a)

level of significance =0.05

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

b)

applying chi square goodness of fit test:

           relative observed Expected residual Chi square
category frequency Oi Ei=total*p R2i=(Oi-Ei)/√Ei R2i=(Oi-Ei)2/Ei
<5 0.072 48 32.76 2.66 7.090
5-14 0.136 68 61.88 0.78 0.605
15-64 0.671 294 305.31 -0.65 0.419
>65 0.121 45 55.06 -1.36 1.836
total 1.000 455 455 9.950

value of the chi-square statistic =9.950

Are all the expected frequencies greater than 5? Yes

chi-square

degree of freedom =categories-1= 3

c)

p value =0.0190

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

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


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