Question

According to the Census Bureau, the distribution by ethnic background of the New York City population in a recent year was

Hispanic: 28% Black: 24% White: 35% Asian: 12% Others: 1%
The manager of a large housing complex in the city wonders whether the distribution byrace of the complex’s residents is consistent with the population distribution. To find out,she records data from random sample of 800 residents. The table below displays the sample data.

Race	Hispanic	Black	White	Asian	Others
Count	212	202	270	94	22

Are these data significantly different from the city’s distribution by race? Carry out anappropriate test at the ? = 0.05 level of significance to support your answer.

a. State the null and alternate hypotheses (write it mathematically) and writeyour claim.

b. Find the standardized test statistic

c. Identify the Rejection region (critical region) and fail to reject region.

4. Decide whether to reject or fail to reject the null.

5. Make an interpretation of your decision in the context.

Answer 1

a.

p ₁₀ , p ₂₀ , p ₃₀ , p ₄₀ , p ₅₀ being the proportion of participants in the population and p ₁ , p ₂ , p ₃ , p ₄ , p ₅

in the sample, we can write our null hypothesis as:

H₀ : p₁ = p ₁₀ , p₂ = p ₂₀ , p₃ = p ₃₀ , p₄ = p₄₀ , p₅ = p₅₀

H₁ : At least one of the values of proportion of participants in the sample has a different value than in the population i.e. is not equal to it's proportion in the population.

b.

Chi- square test statistic also known as X² statistic is given as:

O are the observed frequencies and are clear in the sample data. E are expected frequency and can be calculated using our sample size and population proportion (eg. we expect 28% of 800 i.e. 224 of them to be black):

224, 192, 280, 96, 8 are the expected frequencies.

X² = (212-224)²/224 + (202-192)²/192 + (270-280)²/280 + (94-96)²/96 + (22-8)²/8

X² = 26.06

c.

For ? = 0.05 and df = 4 (i.e. number of cardinals - 1) :

critical value = 9.488

All the test statistic values below 9.488 lie in the fail to reject region and larger values lie in the critical region or rejection region. The statistic of our sample lies in the critical region so we decide to reject the null hypothesis.

The 800 resident sample we tested here was significantly different from city's distribution by race.