The table below shows the age of some participants in a quiz competition and the number of questions they could answer correctly:

Age (years) (x) 15 21 17 22 16 19 18

Number of questions they could answer (y) 17 17 17 17 17 17 17

What is the correlation coefficient for the data, and what does it represent?
0; it represents no correlation between x and y
1; it represents a linear positive correlation between x and y
−1; it represents a linear negative correlation between x and y
1; it represents a linear negative correlation between x and y

0 no correlation

0; it represents no correlation between x and y

To find the correlation coefficient for the given data, you can use a formula called the Pearson correlation coefficient or Pearson's r. The formula is as follows:

r = Σ((xi - x̄)(yi - ȳ)) / (√(Σ(xi - x̄)²) * √(Σ(yi - ȳ)²))

Where:
- xi and yi are the individual data points for x and y respectively.
- x̄ and ȳ are the mean values of x and y respectively.
- Σ represents the sum of the given values.

Let's calculate the correlation coefficient step by step:

1. Calculate the mean of x and y:
x̄ = (15 + 21 + 17 + 22 + 16 + 19 + 18) / 7 = 18
ȳ = (17 + 17 + 17 + 17 + 17 + 17 + 17) / 7 = 17

2. Calculate the deviations from the mean for both x and y:
For x: xi - x̄ = 15 - 18, 21 - 18, 17 - 18, 22 - 18, 16 - 18, 19 - 18, 18 - 18
= -3, 3, -1, 4, -2, 1, 0
For y: yi - ȳ = 17 - 17, 17 - 17, 17 - 17, 17 - 17, 17 - 17, 17 - 17, 17 - 17
= 0, 0, 0, 0, 0, 0, 0

3. Square the deviations from the mean for both x and y:
For x: (-3)², 3², (-1)², 4², (-2)², 1², 0²
= 9, 9, 1, 16, 4, 1, 0
For y: 0², 0², 0², 0², 0², 0², 0²
= 0, 0, 0, 0, 0, 0, 0

4. Calculate the sum of the squared deviations for both x and y:
For x: Σ(xi - x̄)² = 9 + 9 + 1 + 16 + 4 + 1 + 0
= 40
For y: Σ(yi - ȳ)² = 0 + 0 + 0 + 0 + 0 + 0 + 0
= 0

5. Calculate the sum of the product of the deviations from the mean for both x and y:
Σ((xi - x̄)(yi - ȳ)) = (-3 * 0) + (3 * 0) + (-1 * 0) + (4 * 0) + (-2 * 0) + (1 * 0) + (0 * 0)
= 0

6. Finally, calculate the correlation coefficient using the formula:
r = Σ((xi - x̄)(yi - ȳ)) / (√(Σ(xi - x̄)²) * √(Σ(yi - ȳ)²))
= (0) / (√(40) * √(0))
= 0 / (6.325 * 0)
= 0 / 0

Since we have an indeterminate form (0/0), we cannot directly calculate the correlation coefficient for this data. However, we can observe that the values of y remain constant, which means that there is no variation in y. Consequently, there is no linear correlation between x and y in this data set.

So, the correct answer is:
0; it represents no correlation between x and y.

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