Question

he 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%	50
5 to 14	13.6%	78
15 to 64	67.1%	281
65 and older	12.1%	46

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

chi-square binomial

normal

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

The expected values have been calculated where each expected value = (5/100) * Total. Total = 455

Age(years)	%	Expected Value	Observed Value
Under 5	7.2%	(7.2/100) * 455 = 32.760	50
5 to 14	13.6%	(13.6/100) * 455 = 61.880	78
15 to 64	67.1%	(67.1/100) * 455 = 305.305	281
65 and older	12.1%	(12.1/100) * 455 = 55.055	46
Total	100%	455	455

(a) The Level of significance = 0.05

The Hypothesis:

H0: The Distributions are the same.

Ha: The Distributions are different.

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(b) The Test Statistic: The table below gives the calculation of .

Observed	Expected	O-E	(O-E)2	(O-E)2/E
50	32.76	17.24	297.2176	9.07257631
78	61.88	16.12	259.8544	4.19932773
281	305.305	-24.305	590.733025	1.9348947
46	55.055	-9.055	81.993025	1.48929298
455	455		Total	16.696

test = 16.696

YES, all the expected frequencies are greater than 5.

The sampling distribution to be used is the CHI square Distribution

The degrees of freedom, df = n - 1 = 4 - 1 = 3

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(c) The p value: The p value at = 16.696, df = 3, is P value = 0.0008.

Therefore the range in which the p value lies is Option 6: p value < 0.005.

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(d) The Decision: Option 3 - Since p value is , we reject the null Hypothesis.

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(e) The Conclusion: Option 2 - At the 5% level of significance, there is sufficient evidence to conclude that the village population does not fit the general Canadian population.

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