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%	82
15 to 64	67.1%	284
65 and older	12.1%	43

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

YesNo     

What sampling distribution will you use?

chi-squarebinomial     uniformnormalStudent's t

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

(a)

The level of significance is 0.05

The null and alternate hypotheses are

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.)

Expected frequencies are

For Under 5, E1 = 0.072 * 455 = 32.76

For Age 5 to 14, E2 = 0.136 * 455 = 61.88

For Age 15 to 64, E3 = 0.671 * 455 = 305.305

For Age 65 or older, E4 = 0.121 * 455 = 55.055

chi-square statistic = (Oi - Ei)² / Ei

= (46 - 32.76)² / 32.76 + (82 - 61.88)² / 61.88 + (284 - 305.305)² / 305.305 + (43 - 55.055)² / 55.055

= 16.019

Yes, all Expected frequencies (E1, E2, E3 and E4) are greater than 5.

We have used chi-square sampling distribution.

Degrees of freedom = Number of levels - 1 = 4 - 1 = 3

(c)

P-value of the sample test statistic = P(X² > 16.019) = 0.0011

Thus,

P-value < 0.005

(d)

α = 0.05

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.