Students in 7th and 8th grade were asked the types of movies they prefer. The results of the survey are shown in the two-way table.
| | 7th | 8th | Total |
| :----- | :---- | :---- | :------ |
| Drama | 11% | 15% | 26% |
| Action | 25% | 19% | 44% |
| Comedy | 14% | 16% | 30% |
| Total | 50% | 50% | 100% |
Which statements are correct? Check all that apply.
A. More 7th graders were surveyed than 8th graders.
B. The marginal relative frequency for action movies is [tex]$44 \%$[/tex].
C. The joint relative frequency for 8th graders who like comedy movies is [tex]$16 \%$[/tex].
D. Most students prefer drama movies.
E. Half of the 7th graders prefer action movies.
Answer (1)
7th and 8th graders each represent 50% of the survey, disproving the first statement.
The marginal relative frequency for action movies is indeed 44% .
The joint relative frequency for 8th graders liking comedy is 16% .
Action movies are preferred by most students ( 44% ), not drama.
Half the 7th graders prefer action movies, as 50% 25% = 50% . Therefore, the correct statements are: The marginal relative frequency for action movies is 44% , The joint relative frequency for 8th graders who like comedy movies is 16% , and Half of the 7th graders prefer action movies.
Explanation
Analyze the problem We are given a relative frequency table showing movie preferences of 7th and 8th graders. We need to evaluate the correctness of the given statements based on the data in the table.
Evaluate statement 1 The first statement is: "More 7th graders were surveyed than 8th graders." The table shows that 7th graders represent 50% of the total and 8th graders also represent 50% of the total. Therefore, the number of 7th graders surveyed is the same as the number of 8th graders surveyed. This statement is incorrect.
Evaluate statement 2 The second statement is: "The marginal relative frequency for action movies is 44%." Looking at the table, the 'Total' column for 'Action' movies shows 44%. This statement is correct.
Evaluate statement 3 The third statement is: "The joint relative frequency for 8th graders who like comedy movies is 16%." Looking at the table, the intersection of '8th' grade and 'Comedy' movies shows 16%. This statement is correct.
Evaluate statement 4 The fourth statement is: "Most students prefer drama movies." The marginal relative frequencies are: Drama (26%), Action (44%), and Comedy (30%). Since 44% > 30% > 26%, most students prefer action movies, not drama movies. This statement is incorrect.
Evaluate statement 5 The fifth statement is: "Half of the 7th graders prefer action movies." The table shows that 25% of the total students are 7th graders who prefer action movies. Since 7th graders represent 50% of the total students, the proportion of 7th graders who prefer action movies is calculated as: 50% 25% = 0.5 = 50% . Therefore, half of the 7th graders prefer action movies. This statement is correct.
Final Answer The correct statements are:
The marginal relative frequency for action movies is 44%.
The joint relative frequency for 8th graders who like comedy movies is 16%.
Half of the 7th graders prefer action movies.
Examples
Understanding relative frequency tables helps in analyzing survey data. For example, a company can use this to understand customer preferences for different products and target their marketing efforts accordingly. Imagine a school wants to understand what kind of activities students prefer. By creating a relative frequency table, they can quickly see which activities are most popular and allocate resources accordingly, ensuring maximum student engagement.
