Question

a random sample of 250 campers at a summer camp were surveyed to see what water activities they were most interested in participating in. the study found that 120 campers were not interested in paddleboarding. the study also showed that 40 of the campers who were interested in paddleboarding were also interested in waterskiing and 90 of the students were not interested in waterskiing. based on the given information, which of the following relative frequency tables summarizes the data? four tables are presented with rows: waterskiing, no waterskiing, total; columns: paddleboarding, no paddleboarding, total. first table: waterskiing - paddleboarding: 48, no paddleboarding: 112, total: 160; no waterskiing - paddleboarding: 82, no paddleboarding: 8, total: 90; total - paddleboarding: 130, no paddleboarding: 120, total: 250. second table: waterskiing - paddleboarding: 18.2%, no paddleboarding: 44.8%, total: 64%; no waterskiing - paddleboarding: 32.8%, no paddleboarding: 3.2%, total: 36%; total - paddleboarding: 52%, no paddleboarding: 48%, total: 100%. third table: waterskiing - paddleboarding: 48, no paddleboarding: 60, total: 108; no waterskiing - paddleboarding: 82, no paddleboarding: 60, total: 142; total - paddleboarding: 130, no paddleboarding: 120, total: 250. fourth table: waterskiing - paddleboarding: 18.2%, no paddleboarding: 24%; no waterskiing - paddleboarding: 32.8%, no paddleboarding: 24%; total - paddleboarding: 52%, no paddleboarding: 48%, total: 100% (percentages may have formatting variations).

Explanation:

Step1: Analyze the given data

We know the total number of campers is 250. 120 are not interested in paddleboarding, 130 are interested in paddleboarding. 90 are not interested in waterskiing, so 250 - 90 = 160 are interested in waterskiing. Also, 40 of those interested in paddleboarding are interested in waterskiing, so the number of paddleboarding campers not interested in waterskiing is 130 - 40 = 90. The number of non - paddleboarding campers interested in waterskiing is 160 - 40 = 120, and non - paddleboarding campers not interested in waterskiing is 120 - 120 = 0? Wait, no, let's recast in terms of relative frequencies.

Relative frequency is calculated as (frequency)/(total number of observations). The total number of observations is 250.

For the "Paddleboarding" and "Waterskiing" cell: The number of campers interested in both is 40, so relative frequency is $\frac{40}{250}=0.16 = 16\%$? Wait, no, the first table (the fourth one from the left) has Paddleboarding - Waterskiing as 48? Wait, maybe I misread. Wait, the problem says 120 campers not interested in paddleboarding, 130 interested. 90 not interested in waterskiing, so 160 interested. 40 of paddleboarding - interested are waterskiing - interested. So:

- Paddleboarding (130) and Waterskiing: 40, relative frequency $\frac{40}{250}=0.16 = 16\%$? But the third table (from left) has 18.2%? Wait, maybe the correct table is the one where:

Total campers = 250.

- Paddleboarding (130):
- Waterskiing: Let's say the number is $x$, non - waterskiing: $130 - x$
- No Paddleboarding (120):
- Waterskiing: $160 - x$, non - waterskiing: $120-(160 - x)=x - 40$

We also know that 90 are not interested in waterskiing, so $(130 - x)+(x - 40)=90$, which simplifies to 90 = 90, so we need to use the other info: 40 of paddleboarding - interested are waterskiing - interested, so $x = 40$. Then:

- Paddleboarding - Waterskiing: 40, relative frequency $\frac{40}{250}=0.16 = 16\%$? But the first table (fourth from left) has 48? Wait, maybe the numbers are counts first, then relative frequencies. Let's check the tables:

First table (rightmost):

- Paddleboarding - Waterskiing: 48, No Paddleboarding - Waterskiing: 112, total Waterskiing: 160 (48 + 112), which matches 250 - 90 = 160.

- Paddleboarding - No Waterskiing: 82, No Paddleboarding - No Waterskiing: 8, total No Waterskiing: 90 (82+8), which matches.

- Paddleboarding total: 48 + 82 = 130, No Paddleboarding total: 112+8 = 120, total 250.

Now, let's calculate relative frequencies:

- Paddleboarding - Waterskiing: $\frac{48}{250}=0.192 = 19.2\%$

- Paddleboarding - No Waterskiing: $\frac{82}{250}=0.328 = 32.8\%$

- No Paddleboarding - Waterskiing: $\frac{112}{250}=0.448 = 44.8\%$

- No Paddleboarding - No Waterskiing: $\frac{8}{250}=0.032 = 3.2\%$

- Paddleboarding total: $\frac{130}{250}=0.52 = 52\%$

- No Paddleboarding total: $\frac{120}{250}=0.48 = 48\%$

- Waterskiing total: $\frac{160}{250}=0.64 = 64\%$

- No Waterskiing total: $\frac{90}{250}=0.36 = 36\%$

Which matches the third table from the left (the one with 18.2%? Wait, no, 48/250 is 0.192 = 19.2%, 82/250 = 0.328 = 32.8%, 112/250 = 0.448 = 44.8%, 8/250 = 0.032 = 3.2%, 130/250 = 0.52, 120/250 = 0.48, 160/250 = 0.64, 90/250 = 0.36. So the third table (the one with 18.2%? Wait, no, the third table from the left has Paddleboarding - Waterskiing as 18.2% (which is close to 19.2%? Maybe a typo, but the table with relative frequencies where:

- Paddleboarding - Waterskiing: 18.2% (wait, 48/250 is 19.2%, maybe 45/250? No, the correct table is the one where the relative frequencies are calculated correctly. The third table (from the left) has:

Paddleboarding - Waterskiing: 18.2% (which is $\frac{45.5}{250}$? No, 250*0.182 = 45.5, no. Wait, the fourth table (rightmost) is a frequency table, the third is relative. Let's check the third table:

- Paddleboarding - Waterskiing: 18.2% (18.2% of 250 is 45.5, no). Wait, the correct way is:

We know that:

- Number of campers interested in paddleboarding (P) = 130, so relative frequency of P is $\frac{130}{250}=0.52 = 52\%$

- Number of campers not interested in paddleboarding (NP) = 120, relative frequency $\frac{120}{250}=0.48 = 48\%$

- Number of campers interested in waterskiing (W) = 160, relative frequency $\frac{160}{250}=0.64 = 64\%$

- Number of campers not interested in waterskiing (NW) = 90, relative frequency $\frac{90}{250}=0.36 = 36\%$

- Number of campers interested in both P and W: Let's say $n(P\cap W)=40$, relative frequency $\frac{40}{250}=0.16 = 16\%$? But the third table has 18.2% for P and W. Wait, maybe the problem has 45 campers? No, the problem says 40. Wait, maybe I made a mistake in the initial data. Let's look at the tables:

The third table (from the left) has:

- Paddleboarding - Waterskiing: 18.2% (which is $\frac{45.5}{250}$, no, 250*0.182 = 45.5). The fourth table (rightmost) has Paddleboarding - Waterskiing as 48, No Paddleboarding - Waterskiing as 112, total Waterskiing 160 (48 + 112), Paddleboarding - No Waterskiing 82, No Paddleboarding - No Waterskiing 8, total No Waterskiing 90 (82 + 8), Paddleboarding total 130 (48+82), No Paddleboarding total 120 (112 + 8), total 250. Now, calculating relative frequencies:

- Paddleboarding - Waterskiing: $\frac{48}{250}=0.192 = 19.2\%$ (close to 18.2%? Maybe a rounding error, 45/250 = 0.18 = 18%). Wait, the problem says "40 of the campers who were interested in paddleboarding were also interested in waterskiing". So $n(P\cap W)=40$, so relative frequency is $\frac{40}{250}=0.16 = 16\%$? No, 40/250 = 0.16, 16%. But the third table has 18.2%. Wait, maybe the correct table is the third one (from the left) because:

- Paddleboarding - Waterskiing: 18.2% (which is approximately $\frac{45.5}{250}$, no). Wait, the key is that the relative frequency table should have:

- For Paddleboarding (130) and Waterskiing: let's say the number is $x$, relative frequency $\frac{x}{250}$

- Paddleboarding and No Waterskiing: $\frac{130 - x}{250}$

- No Paddleboarding and Waterskiing: $\frac{160 - x}{250}$

- No Paddleboarding and No Waterskiing: $\frac{120-(160 - x)}{250}=\frac{x - 40}{250}$

We know that 90 are not interested in waterskiing, so $\frac{(130 - x)+(x - 40)}{250}=\frac{90}{250}$, which is true.

If we take $x = 45$ (to get 18% approx), but the problem says 40. Wait, maybe the problem has a typo, but the third table (from the left) has the correct relative frequencies:

- Paddleboarding - Waterskiing: 18.2% (which is $\frac{45.5}{250}$, no), 250*0.182 = 45.5. Alternatively, the third table's relative frequencies add up: 18.2%+32.8% = 51%? No, 18.2+32.8 = 51, close to 52. 44.8+3.2 = 48, close to 48. 18.2+44.8 = 63, close to 64. 32.8+3.2 = 36. So the third table (from the left) is the correct relative frequency table.

Step2: Identify the correct table

The third table (from the left) has the correct relative frequencies:

- Paddleboarding - Waterskiing: 18.2%

- Paddleboarding - No Waterskiing: 32.8% (since 52% - 18.2% = 33.8%? No, 18.2+32.8 = 51, close to 52. 44.8+3.2 = 48, close to 48. 18.2+44.8 = 63, close to 64. 32.8+3.2 = 36. So the third table (the one with 18.2%, 32.8%, 44.8%, 3.2%, 52%, 48%, 64%, 36%, 100%) is the correct relative frequency table.

Answer:

The third table from the left (the one with relative frequencies: Paddleboarding - Waterskiing: 18.2%, Paddleboarding - No Waterskiing: 32.8%, No Paddleboarding - Waterskiing: 44.8%, No Paddleboarding - No Waterskiing: 3.2%, and totals 52%, 48%, 64%, 36%, 100%)