Question

mr. capuano, an art teacher, surveyed his students to find out whether they are satisfied with his classes. he also noted which class each student had taken. oil-painting satisfied: 17 not satisfied: 3 sculpture satisfied: 25 not satisfied: 5 art classes table with columns oil painting, sculpture, total; row satisfied: 34%, (blank), 84% what is the value of x in the relative frequency table for the survey results? round the answer to the nearest percent. options: 3%, 6%, 8%, 15%

Explanation:

Step1: Find total satisfied students

First, we find the total number of satisfied students from the raw data. For oil - painting, satisfied is 17, and for sculpture, satisfied is 25. So total satisfied students \(= 17 + 25=42\).

Step2: Find the relative frequency of satisfied sculpture students

We know that the relative frequency of satisfied oil - painting students is 34% or 0.34, and the number of satisfied oil - painting students is 17. Let the total number of students be \(N\). We know that relative frequency \(=\frac{\text{number of cases}}{\text{total number of cases}}\), so for oil - painting satisfied: \(0.34=\frac{17}{N}\), solving for \(N\) we get \(N = \frac{17}{0.34}=50\).

Now, the number of satisfied sculpture students is 25, and the total number of students is 50. So the relative frequency of satisfied sculpture students is \(\frac{25}{50}=0.5 = 50\%\)? Wait, no, wait the table has "Satisfied" row with oil - painting 34% and total 84%. Wait, maybe we should use the table's total for satisfied. The total relative frequency of satisfied students is 84%. The relative frequency of satisfied oil - painting is 34%. So the relative frequency of satisfied sculpture (\(x\)) is \(84\% - 34\%=50\%\)? No, that can't be. Wait, maybe I misread. Wait the raw data: oil - painting satisfied 17, not satisfied 3; sculpture satisfied 25, not satisfied 5. Total students: \(17 + 3+25 + 5=50\).

Satisfied total: \(17 + 25 = 42\), relative frequency of satisfied: \(\frac{42}{50}=0.84 = 84\%\), which matches the table.

Relative frequency of satisfied oil - painting: \(\frac{17}{50}=0.34 = 34\%\), which matches the table.

So relative frequency of satisfied sculpture: \(\frac{25}{50}=0.5 = 50\%\)? But that's not one of the options. Wait, maybe the question is about the "Not satisfied" row? Wait the table is cut off. Wait the options are 3%, 6%, 8%, 15%. Wait maybe the \(x\) is the relative frequency of "Not satisfied" for one of the classes.

Let's recalculate. Total not satisfied: \(3 + 5=8\). Total students: 50.

Relative frequency of not satisfied oil - painting: \(\frac{3}{50}=0.06 = 6\%\). Wait, maybe that's \(x\). Let's check:

Total not satisfied relative frequency: \(\frac{8}{50}=0.16 = 16\%\). If oil - painting not satisfied is 6% (3/50), then sculpture not satisfied is \(16\% - 6\% = 10\%\), but that's not an option. Wait, maybe the table is structured as:

Rows: Satisfied, Not Satisfied

Columns: Oil Painting, Sculpture, Total

We know Satisfied: Oil Painting 34% (17/50), Total 84% (42/50), so Sculpture Satisfied is \(84\% - 34\% = 50\%\) (25/50).

Now Not Satisfied: Oil Painting: 3/50 = 6%, Sculpture: 5/50 = 10%, Total: 8/50 = 16%.

Ah! Maybe the \(x\) is the relative frequency of "Not satisfied" for Oil Painting? Let's check the options. 6% is an option.

Let's verify:

Number of not satisfied in oil - painting: 3. Total number of students: \(17 + 3+25 + 5 = 50\).

Relative frequency \(=\frac{3}{50}=0.06 = 6\%\).

Step3: Confirm the calculation

Relative frequency is calculated as \(\text{Relative Frequency}=\frac{\text{Number of cases}}{\text{Total number of cases}}\). For not - satisfied oil - painting students, number of cases is 3, total number of students is \(17 + 3+25 + 5=50\). So \(\frac{3}{50}=0.06 = 6\%\).

Answer:

6% (the option corresponding to 6%)