Question

The accuracy of a census report on a city in southern California was questioned by some government officials. A random sample of 1215 people living in the city was used to check the report, and the results are shown below.

Ethnic Origin	Census Percent	Sample Result
Black	10%	126
Asian	3%	39
Anglo	38%	484
Latino/Latina	41%	493
Native American	6%	62
All others	2%	11

Using a 1% level of significance, test the claim that the census distribution and the sample distribution agree.

(a) What is the level of significance?
_______________

State the null and alternate hypotheses.

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

(b) Find the value of the chi-square statistic for the sample. (Round the expected frequencies to at least three decimal places. Round the test statistic to three decimal places.)
______________

Are all the expected frequencies greater than 5?

Yes

No    

What sampling distribution will you use?

chi-square

uniform    

binomial

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 1% level of significance, the evidence is sufficient to conclude that census distribution and the ethnic origin distribution of city residents are different.

At the 1% level of significance, the evidence is insufficient to conclude that census distribution and the ethnic origin distribution of city residents are different.    

Answer 1

a)
Significance level = 0.01

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

b)

Ethnic Origin	Expected Percentage	Observed Number	Expected Number	(Oi - Ei)^2/Ei
Black	10%	126	121.500	0.167
Asian	3%	39	36.450	0.178
Anglo	38%	484	461.700	1.077
Latino	41%	493	498.150	0.053
Native American	6%	62	72.900	1.630
All Others	2%	11	24.300	7.279
		1215		10.384

test statistic, chi-square = sum((Oi - Ei)^2/Ei) = 10.384

All expected frequencies are greater than 5 - Yes

sampling distribution - chi-square

degrees of freedom = 6 - 1 = 5

c)
0.050 < P-value < 0.100

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

e)
At the 1% level of significance, the evidence is insufficient to conclude that census distribution and the ethnic origin distribution of city residents are different.