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%	46
5 to 14	13.6%	70
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 different.
H₁: The distributions are the same.H₀: The distributions are the same.
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 different.

(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?

YesNo    

What sampling distribution will you use?

uniformnormal    chi-squareStudent's tbinomial

What are the degrees of freedom?


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

P-value > 0.1000.050 < P-value < 0.100    0.025 < P-value < 0.0500.010 < P-value < 0.0250.005 < P-value < 0.010P-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

Solution:

Given: 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%	46
5 to 14	13.6%	70
15 to 64	67.1%	294
65 and older	12.1%	45

Part (a) What is the level of significance?

the level of significance = 5% = 0.05

State the null and alternate hypotheses.

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

Part (b) Find the value of the chi-square statistic for the sample.

Chi square test statistic for goodness of fit

Where

O_i = Observed Counts

E_i =Expected Counts

To get E_i values , Multiply % values by 455

Thus we need to make following table:

Age (years)	Percent of Canadian Population	Oi: Observed Number in the Village	Ei: Expected Frequency	Oi^2/Ei
Under 5	7.20%	46	32.76	64.591
5 to 14	13.60%	70	61.88	79.186
15 to 64	67.10%	294	305.305	283.114
65 and older	12.10%	45	55.055	36.781
		455

Thus

Are all the expected frequencies greater than 5?

Yes

What sampling distribution will you use?

chi-square

What are the degrees of freedom?

df = k - 1= 4 - 1 = 3

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

df = 3 ,

Thus look in Chi-square table for df = 3 row and find interval in which fall then find corresponding right tail area interval.

8.671 is between 7.815 and 9.348

Corresponding right tail area interval is 0.025 to 0.050

thus 0.025 < p-value < 0.050

Thus correct option is:  0.025 < P-value < 0.050

Part (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?

0.025 < P-value < 0.050 , that means P-value < 0.05

thus we reject H0.

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

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

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