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%	139
Asian	3%	35
Anglo	38%	469
Latino/Latina	41%	503
Native American	6%	56
All others	2%	13

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 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 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?

YesNo    

What sampling distribution will you use?

uniformnormal    binomialchi-squareStudent'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 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) level of significance =0.01

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

b)

Applying chi square test:

	relative	observed	Expected	residual	Chi square
category	frequency	Oi	Ei=total*p	R2i=(Oi-Ei)/√Ei	R2i=(Oi-Ei)2/Ei
Black	0.10	139.00	121.50	1.59	2.521
Asian	0.03	35.00	36.45	-0.24	0.058
Anglo	0.38	469.00	461.70	0.34	0.115
Latino/Latina	0.41	503.00	498.15	0.22	0.047
Native American	0.06	56.00	72.90	-1.98	3.918
All others	0.02	13.00	24.30	-2.29	5.255
total	1.000	1215	1215		11.913

value of the chi-square statistic =11.913

Are all the expected frequencies greater than 5? Yes

What sampling distribution will you use? Chi square

degrees of freedom =5

c) 0.025 < P-value < 0.05

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.