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	58	12	322	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.

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 = __

Answer 1

Solution:

The given data is as shown below.

Ethnicity	Observed frequency(Oi)
African-American	58
Asian	12
Caucasian	322
Hispanic	6

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

Step 1:State the null and alternative hypothesis.

Null hypothesis:Ho:p₁=0.177,p₂=0.032 ,p₃=0.734 and p₄=0.057

and Alternative hypothesis:Ha: Atleast one of the population proportions is not equal to its corresponding hypothesized proportion.

Step 2:Degree of freedom and level of significance

DF=k-1=4-1=3 where DF is the degrees of freedom, k is the number of levels of the categorical variable

and level of significance,=0.01

Step 3:Calculation of test statistics

The test statistics,X²=(Oi-Ei)²/Ei

The below table shows the values required to calculate the test statistics.

Ethnicity	Observed frequency(Oi)	Proportions	Expected frequency(Ei)	(Oi-Ei)	(Oi-Ei)2	(Oi-Ei)2/Ei
African-American	58	0.177	398*0.177=70.446	-12.446	154.9029	2.199
Asian	12	0.032	398*0.032=12.736	-0.736	0.541696	0.043
Caucasian	322	0.734	398*0.734=292.132	29.868	892.0974	3.054
Hispanic	6	0.057	398*0.057=22.686	-16.686	278.4226	12.273
Total	398		398			17.568

Substituting the values form the table in the formula,X²=(Oi-Ei)²/Ei

=17.568

   X²=17.57(round to 2 decimal places)

Step 4:Determination of X² critical value.

The Chi-square distribution table is as shown below.

For DF=3 and =0.01, the X² critical value=11.345

Step 5:Conclusion

Since X² observed value=17.5 which is greater than the X² critical value=11.345, we reject the null hypothesis and conclude that atleast one of the population proportions is not equal to its corresponding hypothesized proportion.

Calculation of P-value.

The P-value is 0.0005.