Question

a group of 55 football players was polled about which synthetic grass they like to play on, chemgrass or chemstrand. the data from the survey is shown in the venn diagram. use the venn diagram to find the missing values in the frequency table.
synthetic grass preference
venn diagram: chemstrand circle has 15, overlap has 13, chemgrass circle has 17, outside both has 10
synthetic grass preference
frequency table with columns: likes chemstrand, does not like chemstrand, total
what are the values of x, y, and z?

• x = 17, y = 15, and z = 13
• x = 13, y = 15, and z = 17
• x = 15, y = 13, and z = 17
• x = 13, y = 17, and z = 15

Explanation:

Step1: Analyze Venn Diagram for x

x represents the number of players who like Chemgrass (only Chemgrass + both). From Venn, only Chemgrass is 17, both is 13? Wait, no, wait. Wait, the Venn: Chemstrand circle has 15 (only Chemstrand), overlap 13 (both), Chemgrass circle has 17 (only Chemgrass), and outside both is 10. Wait, total players: 15 + 13 + 17 + 10 = 55, which matches. Now, the frequency table: let's assume the rows are "Likes Chemgrass" and "Does Not Like Chemgrass", columns "Likes Chemstrand", "Does Not Like Chemstrand", "Total". Wait, maybe x is "Likes Chemgrass" total? Wait, no, let's re-express. Wait, the options: x, y, z. Let's think:

Wait, the Venn:
- Only Chemstrand: 15
- Both: 13
- Only Chemgrass: 17
- Neither: 10

Now, for the frequency table (assuming rows: Likes Chemgrass, Does Not Like Chemgrass; columns: Likes Chemstrand, Does Not Like Chemstrand, Total).

First, "Likes Chemgrass" row:
- Likes Chemstrand (both): 13
- Does Not Like Chemstrand (only Chemgrass): 17
- Total (13 + 17) = 30? Wait, no, maybe the columns are Likes Chemstrand, Does Not Like Chemstrand, and rows are Likes Chemgrass, Does Not Like Chemgrass. Wait, maybe x is "Likes Chemgrass" total? Wait, no, let's check the options. The options have x, y, z. Let's see:

Wait, another approach: The total number of players who like Chemstrand is only Chemstrand + both = 15 + 13 = 28. The total number of players who do not like Chemstrand is only Chemgrass + neither = 17 + 10 = 27. Now, for the "Likes Chemgrass" row: Likes Chemstrand (both) is 13, Does Not Like Chemstrand (only Chemgrass) is 17, so total for "Likes Chemgrass" is 13 + 17 = 30? No, wait, maybe the table is:

| | Likes Chemstrand | Does Not Like Chemstrand | Total |
|----------------|------------------|--------------------------|-------|
| Likes Chemgrass| x | y | z |
| Does Not... |... |... |... |

Wait, no, maybe the rows are "Likes Chemstrand" and "Does Not Like Chemstrand", columns "Likes Chemgrass", "Does Not Like Chemgrass", "Total". Wait, the original problem's table is partially visible, but from the Venn:

- Likes Chemstrand: only Chemstrand (15) + both (13) = 28
- Does Not Like Chemstrand: only Chemgrass (17) + neither (10) = 27
- Likes Chemgrass: both (13) + only Chemgrass (17) = 30
- Does Not Like Chemgrass: only Chemstrand (15) + neither (10) = 25

Wait, but the options: let's check the options. The options are:

1. x=17, y=15, z=13 → No
2. x=13, y=15, z=17 → No
3. x=15, y=13, z=17 → No
4. x=13, y=17, z=15 → No, wait, no, wait I must have messed up. Wait, maybe the table is:

Wait, maybe the rows are "Likes Chemgrass" and "Does Not Like Chemgrass", and columns "Likes Chemstrand", "Does Not Like Chemstrand", "Total".

So:

- Likes Chemgrass:
- Likes Chemstrand: both (13) → x?
- Does Not Like Chemstrand: only Chemgrass (17) → y?
- Total: 13 + 17 = 30 → z? No, that doesn't match options. Wait, no, maybe the other way. Wait, the options have x, y, z as 13,15,17 etc. Wait, let's re-express the Venn:

Only Chemstrand: 15 (so Does Not Like Chemgrass, Likes Chemstrand: 15)

Both: 13 (Likes Chemgrass, Likes Chemstrand: 13)

Only Chemgrass: 17 (Likes Chemgrass, Does Not Like Chemstrand: 17)

Neither: 10 (Does Not Like Chemgrass, Does Not Like Chemstrand: 10)

Now, the frequency table (assuming columns: Likes Chemstrand, Does Not Like Chemstrand, Total; rows: Likes Chemgrass, Does Not Like Chemgrass):

- Likes Chemgrass row:
- Likes Chemstrand: 13 (both)
- Does Not Like Chemstrand: 17 (only Chemgrass)
- Total: 13 + 17 = 30
- Does Not Like Chemgrass row:
- Likes Chemstrand: 15 (only Chemstrand)
- Does Not Like Chemstrand: 10 (neither)
- Total: 15 + 10 = 25
- Column Totals:
- Likes Chemstrand: 13 + 15 = 28
- Does Not Like Chemstrand: 17 + 10 = 27
- Grand Total: 28 + 27 = 55 (matches)

But the options are x, y, z. Wait, maybe the table is different. Wait, the options are:

Option 2: x=13, y=15, z=17? No, wait, maybe I got x, y, z wrong. Wait, maybe x is "Likes Chemgrass" and "Likes Chemstrand" (both) → 13, y is "Does Not Like Chemgrass" and "Likes Chemstrand" (only Chemstrand) → 15, z is "Likes Chemgrass" and "Does Not Like Chemstrand" (only Chemgrass) → 17? Wait, that would be x=13, y=15, z=17? Wait, no, the second option is x=13, y=15, z=17? Wait, the options:

1. x=17, y=15, z=13 → No
2. x=13, y=15, z=17 → Yes? Wait, no, wait:

Wait, if x is the number of players who like both (13), y is the number who like only Chemstrand (15), z is the number who like only Chemgrass (17). Then x=13, y=15, z=17, which is option 2.

Yes, that makes sense. So:

- x: both (13)
- y: only Chemstrand (15)
- z: only Chemgrass (17)

So the correct option is x=13, y=15, and z=17.

Step2: Confirm with Total

Total players: 13 (both) + 15 (only Chemstrand) + 17 (only Chemgrass) + 10 (neither) = 55, which matches. So the values are x=13, y=15, z=17.

Answer:

B. x = 13, y = 15, and z = 17