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%	120
Asian	3%	47
Anglo	38%	469
Latino/Latina	41%	503
Native American	6%	63
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 the same.    

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

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

(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?

Student's t

chi-square    

binomial

uniform

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 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

Solution:

Given:

. 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%	120
Asian	3%	47
Anglo	38%	469
Latino/Latina	41%	503
Native American	6%	63
All others	2%	13

Part a) What is the level of significance?

the level of significance =

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.

To get E_i , multiply % values by N = 1215

Ethnic Origin	Census Percent	Sample Result	Ei	Oi^2/Ei
Black	10%	120	121.5	118.519
Asian	3%	47	36.45	60.604
Anglo	38%	469	461.7	476.415
Latino/Latina	41%	503	498.15	507.897
Native American	6%	63	72.9	54.444
All others	2%	13	24.3	6.955
		N = 1215

Thus

Are all the expected frequencies greater than 5?

Yes

What sampling distribution will you use?

chi-square  

What are the degrees of freedom?

the degrees of freedom = k - 1

the degrees of freedom = 6 - 1

the degrees of freedom = 5

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

look in Chi-square table for df = 5 row and find the interval in which fall. then find corresponding one tail area interval.

fall between 9.236 and 11.070

Corresponding area is between 0.050 and 0.100

Thus range of p-values is:
0.050 < p-value < 0.100

Part d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis that

Since 0.050 < p-value < 0.100, it means that p-value > 0.01 significance level, we fail to reject H0.

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

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

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.