Question

a random sample of 200 high school students were surveyed to see whether they were enrolled in advanced algebra and physics. the study found that 116 of the sample were not enrolled in advanced algebra. the study also found that 84 of the students enrolled in advanced algebra were also enrolled in physics and 160 of the students were not enrolled in physics. based on the given information, which of the following relative frequency tables summarizes the data? (tables are shown with algebra/not in algebra rows and physics/not in physics columns with various percentages and counts)

Explanation:

Step1: Identify given values

Total students \( n = 200 \).
Enrolled in advanced algebra (Algebra): \( 200 - 116 = 84 \).
Enrolled in physics (Physics): \( 200 - 160 = 40 \).
Enrolled in both Algebra and Physics: Let's find. From the first table (let's check counts):
Algebra (84) = Physics (40) + Not in Physics (Algebra). Wait, first table: Algebra row, Physics column is 36, Not in Algebra row, Physics column is 4? Wait no, first table: Physics row, Algebra: 36, Not in Algebra: 4 (total 40). Not in Physics row, Algebra: 48, Not in Algebra: 112 (total 160). Total Algebra: 36 + 48 = 84, Total Not in Algebra: 4 + 112 = 116. Total Physics: 36 + 4 = 40, Total Not in Physics: 48 + 112 = 160. Now check relative frequencies (percentages):

• Algebra and Physics: \( \frac{36}{200} \times 100 = 18\% \)
• Algebra and Not in Physics: \( \frac{48}{200} \times 100 = 24\% \)
• Not in Algebra and Physics: \( \frac{4}{200} \times 100 = 2\% \)
• Not in Algebra and Not in Physics: \( \frac{112}{200} \times 100 = 56\% \)? Wait no, first table's totals: Physics row total 40 (36+4), Not in Physics row total 160 (48+112). Algebra column total 84 (36+48), Not in Algebra column total 116 (4+112). Now check the second table (percentages):

Algebra (84) as % of 200: \( \frac{84}{200} = 42\% \), Not in Algebra: \( \frac{116}{200} = 58\% \).
Physics (40) as % of 200: \( \frac{40}{200} = 20\% \), Not in Physics: \( \frac{160}{200} = 80\% \).
Now, Algebra and Physics: \( \frac{36}{200} = 18\% \), Algebra and Not in Physics: \( \frac{48}{200} = 24\% \), Not in Algebra and Physics: \( \frac{4}{200} = 2\% \), Not in Algebra and Not in Physics: \( \frac{112}{200} = 56\% \)? Wait the second table (second from left) has:
Physics row: Algebra 18%, Not in Algebra 2% (total 20%).
Not in Physics row: Algebra 24%, Not in Algebra 56% (total 80%).
Total Algebra: 18% + 24% = 42%, Total Not in Algebra: 2% + 56% = 58%.
Yes, that matches. So the second table (second from left) has correct relative frequencies:

• Physics & Algebra: \( \frac{36}{200} \times 100 = 18\% \)
• Physics & Not in Algebra: \( \frac{4}{200} \times 100 = 2\% \) (total 20% for Physics)
• Not in Physics & Algebra: \( \frac{48}{200} \times 100 = 24\% \)
• Not in Physics & Not in Algebra: \( \frac{112}{200} \times 100 = 56\% \) (total 80% for Not in Physics)
• Total Algebra: 18% + 24% = 42%, Total Not in Algebra: 2% + 56% = 58% (matches \( \frac{84}{200}=42\%, \frac{116}{200}=58\% \))

Step2: Match with tables

The second table (from left, second one) has:
Physics: Algebra 18%, Not in Algebra 2% (total 20%)
Not in Physics: Algebra 24%, Not in Algebra 58%? Wait no, wait the second table (second from left) shows:
Physics row: Algebra 18%, Not in Algebra 2% (total 20%)
Not in Physics row: Algebra 24%, Not in Algebra 56%? Wait no, the second table's Not in Physics row, Not in Algebra is 58%? Wait no, let's recalculate \( \frac{112}{200} = 56\% \). Wait maybe I misread. Wait the second table (second from left) has:
Total Algebra: 42% (18% + 24%), Total Not in Algebra: 58% (2% + 56%). Wait the second table (second from left) is:
Physics: Algebra 18%, Not in Algebra 2% (total 20%)
Not in Physics: Algebra 24%, Not in Algebra 56% (total 80%)
Total Algebra: 18+24=42%, Total Not in Algebra: 2+56=58% (correct, 84/200=42%, 116/200=58%). So this table matches the relative frequencies.

Answer:

The second table (from the left, the one with Physics: 18%, 2%; Not in Physics: 24%, 56%; Total: 42%, 58%)