Question

The authors of a paper concerned about racial stereotypes in television counted the number of times that characters of different ethnicities appeared in commercials aired on a certain city's television stations, resulting in the data in the accompanying table.

Ethnicity	African- American	Asian	Caucasian	Hispanic
Observed Frequency	56	12	320	6

Based on the 2000 Census, the proportion of the U.S. population falling into each of these four ethnic groups are 0.177 for African-American, 0.032 for Asian, 0.734 for Caucasian, and 0.057 for Hispanic. Do the data provide sufficient evidence to conclude that the proportions appearing in commercials are not the same as the census proportions? Test the relevant hypotheses using a significance level of 0.01.

Let p₁, p₂, p₃, and p₄ be the proportions of appearances of the four ethnicities across all commercials.

State the null and alternative hypotheses.

H₀: p₁ = p₂ = p₃ = p₄ = 0.177
H_a: H₀ is not true.

H₀: p₁ = p₂ = p₃ = p₄ = 69.738
H_a: H₀ is not true.    

H₀: p₁ = 0.177, p₂ = 0.032, p₃ = 0.734, p₄ = 0.057
H_a: H₀ is not true.

H₀: p₁ = p₂ = p₃ = p₄ = 56
H_a: H₀ is not true.

H₀: p₁ = 69.738, p₂ = 12.608, p₃ = 289.196, p₄ = 22.458
H_a: H₀ is not true.

Calculate the test statistic. (Round your answer to two decimal places.)
χ² =  

What is the P-value for the test? (Round your answer to four decimal places.)
P-value =  

What can you conclude?

Do not reject H₀. There is convincing evidence to conclude that the proportions of appearances in commercials are not the same as the census proportions.

Reject H₀. There is not enough evidence to conclude that the proportions of appearances in commercials are not the same as the census proportions.     

Reject H₀. There is convincing evidence to conclude that the proportions of appearances in commercials are not the same as the census proportions.

Do not reject H₀. There is not enough evidence to conclude that the proportions of appearances in commercials are not the same as the census proportions.

Answer 1

State the null and alternative hypotheses.

H₀: p₁ = 0.177, p₂ = 0.032, p₃ = 0.734, p₄ = 0.057
H_a: H₀ is not true.

Test Statistic

O : Observed Frequency

E : Expected Frequency

Ethnicity	O: Observed Frequency	Proportion	E: Expected Frequency = Total * Proportion
African-American	56	0.177	394*0.177=69.738
Asian	12	0.032	394*0032=12.608
Caucasian	320	0.734	394*0.734=289.196
Hispanic	6	0.057	394*0.057=22.458
Total	394

O: Observed Frequency	E: Expected Frequency	O-E	(O-E)2	(O-E)2/E
56	69.738	-13.738	188.7326	2.7063
12	12.608	-0.608	0.3697	0.0293
320	289.196	30.804	948.8864	3.2811
6	22.458	-16.458	270.8658	12.0610
			Total	18.0777

Test Statistic

Degrees of freedom = number of ethnicities -1 = 4-1 =3

For 3 degrees of freedom,

p-value = 0.0004

As p-value : 0.0004 < significance level : 0.01 ; Reject the null hypothesis.

Conclusion:

Reject H₀. There is convincing evidence to conclude that the proportions of appearances in commercials are not the same as the census proportions.