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The authors of the paper 鈥淩acial Stereotypes in Children's Television Commercials鈥� (Journal of Advertising Research [2008]: 80-93) counted the number of times that characters of different ethnicities appeared in commercials aired on Philadelphia television stations, resulting in the data in the accompanying table. $$ \begin{array}{l|cccc} \hline \text { Ethnicity } & \begin{array}{l} \text { African- } \\ \text { American } \end{array} & \text { Asian } & \text { Caucasian } & \text { Hispanic } \\ \hline \begin{array}{c} \text { Observed } \\ \text { Frequency } \end{array} & 57 & 11 & 330 & 6 \\ \hline \end{array} $$ Based on the 2000 Census, the proportion of the U.S. population falling into each of these four ethnic groups are \(.177\) for African-American, \(.032\) for Asian, \(.734\) for Caucasian, and \(.057\) 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 .01.

Step by step solution

01

Calculate Expected Frequencies

The expected frequency can be calculated using the formula: \( E_i = n*P_i \) where \( E_i \) is the expected frequency, \( n \) is the total number of observations and \( P_i \) is the proportion in the population for category i. The total number of observations is the sum of the observed frequencies, which is \( 57+11+330+6 = 404 \). Therefore, the expected frequencies are: \( E1 = 404*.177 = 71.51 \) \( E2 = 404*.032 = 12.93 \) \( E3 = 404*.734 = 296.34 \) \( E4 = 404*.057 = 23.03 \)

02

Calculate Chi-Square Test Statistic

The Chi-Square test statistic can be calculated using the formula: \( X^2 = 危 [ (O_i - E_i)^2 / E_i ] \) where \( O_i \) is the observed frequency and \( E_i \) is the expected frequency for category i. The calculated value of X^2 is: \( X^2 = (57 - 71.51)^2 / 71.51 + (11 - 12.93)^2 / 12.93 + (330 - 296.34)^2 / 296.34 + (6 - 23.03)^2 / 23.03 = 2.99 + 0.29 + 3.58 + 12.66 = 19.52 \)

03

Determine Critical Value and Make Decision

The critical value can be found using Chi-Square distribution table. With a significance level of .01 and df = 4-1 = 3, the critical value is 11.34. Since the calculated test statistic 19.52 is greater than the critical value, the null hypothesis is rejected. This indicates that there is significant evidence to conclude that the proportions appearing in commercials are not the same as the census proportions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chi-Square Test

The Chi-Square Test is a statistical method used to determine if there is a significant difference between observed and expected frequencies in categorical data. This test helps evaluate whether differences between categories arise from chance or some specific factors.
It is often used when dealing with population proportions to check if the witnessed data align with expected ratios. In this particular context, we use the Chi-Square Test to compare the proportions of ethnic groups in TV commercials against census data.
To conduct the test, follow these steps:

• Calculate expected frequencies for each category based on known population proportions and total observations.
• Use the formula \( X^2 = \sum \left( \frac{(O_i - E_i)^2}{E_i} \right) \), where \(O_i\) are observed frequencies and \(E_i\) are expected frequencies. This gives the Chi-Square statistic.
• Find the critical value using the Chi-Square distribution table, with degrees of freedom \( df = \,\text{(number of categories)} - 1 \) and the specified significance level.

If the calculated statistic exceeds the critical value, you reject the null hypothesis, suggesting a significant difference between what is observed and what is expected. This means the TV commercial proportions deviate significantly from the census proportions.

Observed Frequencies

Observed Frequencies refer to the actual counts of occurrences that you find in your sample data. In our exercise, these are the numbers representing how often characters of different ethnic backgrounds appear in TV commercials.
These raw data points are crucial because they offer a real-world sample that we can analyze. Here, we have 57 African-American, 11 Asian, 330 Caucasian, and 6 Hispanic appearances.
The observed frequencies provide the basis for analysis, as they are compared against expected frequencies derived from another data source鈥攊n this case, census data鈥攗sing the Chi-Square Test.
The goal is to assess how well these observed frequencies align with population-based expectations. Discrepancies may indicate a deviation influenced by factors such as societal bias or under-representation, warranting further examination.

Census Proportions

The term 'Census Proportions' refers to the distribution of certain demographic characteristics as recorded in a census, typically conducted by a governmental entity. These proportions are used as benchmarks for expected distributions in various studies.
In our scenario, the proportions for different ethnic groups are drawn from the 2000 U.S. Census data:

• African-American: 17.7%
• Asian: 3.2%
• Caucasian: 73.4%
• Hispanic: 5.7%

These percentages form the basis for calculating expected frequencies in the Chi-Square Test. By applying these proportions to the total number of observed characters (404), we establish a baseline for comparison.
Accurate census proportions are critical because they provide a scientifically collected standard reference. Any derivation in commercial representations implies potential disparities that might prompt broader societal discussions or actions.