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%	121
Asian	3%	47
Anglo	38%	471
Latino/Latina	41%	504
Native American	6%	59
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 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 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?

Student's tnormal    uniformbinomialchi-square

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

Solution:

We have to test the claim that  the census distribution and the sample distribution agree.

Part a) What is the level of significance?

The level of significance = 1% = 0.01

State the null and alternate hypotheses.

H₀: The distributions are the same.

Vs

H₁: The distributions are different

Part b) Find the value of the chi-square statistic for the sample.

where

O_i = Observed Counts

E_i =Expected Counts = N * Census Percent

Thus we need to make following table:

Ethnic Origin	Census Percent	Oi : Sample Result	Ei : Expected Frequencies	Oi2 / Ei
Black	10%	121	121.500	120.50206
Asian	3%	47	36.450	60.60357
Anglo	38%	471	461.700	480.48733
Latino/Latina	41%	504	498.150	509.91870
Native American	6%	59	72.900	47.75034
All others	2%	13	24.300	6.95473
		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?

df = k - 1

We have k = 6 Ethnic Origin, thus

df = 6 - 1

df = 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 Chi-square test statistic value = fall.

Then find corresponding right tail area, which would be P-value interval.

fall between 11.070 and 12.833 and these values corresponds to area 0.050 and 0.025 respectively

Thus range of P-values is ( 0.025 , 0.050)

thus we get: 0.025 < P-value < 0.050

Part 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 0.025 < P-value < 0.050, which implies P-value > 0.01 level of significance, thus we fail to reject H0.

Thus correct option is:

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.