The graph below plots the values of y for different values of x:
plot the ordered pairs 1, 1 and 2, 4 and 3, 7 and 4, 9 and 5, 12 and 6, 16
Which correlation coefficient best matches the data plotted on the graph?
A. −0.5
B. 0
C. 0.25
D. 0.90
Is it D. 0.90??
I get 0.90 as the best match.
Ms. Sue can you please help? My home work needs to be turned in in 10 minutes???
To determine the correlation coefficient, we need to consider the relationship between the x and y values in the given ordered pairs. Let's plot the ordered pairs on the graph:
(1, 1), (2, 4), (3, 7), (4, 9), (5, 12), (6, 16)
Based on the plotted points, we can observe that as the x values increase, the corresponding y values also increase. This indicates a positive correlation between x and y.
To calculate the correlation coefficient, we can use the formula:
r = (Σ((x - x̄)(y - ȳ))) / (sqrt(Σ(x - x̄)²) * sqrt(Σ(y - ȳ)²))
Let's calculate the correlation coefficient step-by-step:
1. Calculate the mean (average) of x and y values:
x̄ = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5
ȳ = (1 + 4 + 7 + 9 + 12 + 16) / 6 = 8.2
2. Calculate the deviations from the mean for x and y values:
(x - x̄) = (-2.5, -1.5, -0.5, 0.5, 1.5, 2.5)
(y - ȳ) = (-7.2, -4.2, -1.2, 0.8, 3.8, 7.8)
3. Calculate the products of the deviations for each ordered pair:
(x - x̄)(y - ȳ) = (18.0, 6.3, 0.6, 0.4, 5.7, 19.5)
4. Calculate the sums:
Σ(x - x̄)(y - ȳ) = 51.5
Σ(x - x̄)² = 17.5
Σ(y - ȳ)² = 154.0
5. Calculate the correlation coefficient:
r = 51.5 / (sqrt(17.5) * sqrt(154.0)) ≈ 0.912
The correlation coefficient is approximately 0.912, which is close to 0.90.
Therefore, the most accurate match for the correlation coefficient based on the given data plotted on the graph is D. 0.90.
To determine the correlation coefficient that best matches the data plotted on the graph, we need to find the relationship between the x-values and the y-values.
Given the ordered pairs [(1, 1), (2, 4), (3, 7), (4, 9), (5, 12), (6, 16)], we can plot these points on a graph to visualize the relationship.
By observing the graph, we can see that as the x-values increase, the y-values also increase. The relationship seems to be a positive linear relationship as the points form a straight line that slopes upwards.
Now, we can calculate the correlation coefficient to verify our observation. The correlation coefficient is a measure of the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where -1 indicates a strong negative linear relationship, 1 indicates a strong positive linear relationship, and 0 indicates no linear relationship.
To calculate the correlation coefficient, we can use the formula:
r = (n * ∑xy - ∑x * ∑y) / sqrt((n * ∑x^2 - (∑x)^2) * (n * ∑y^2 - (∑y)^2))
where n is the number of data points, ∑xy is the sum of the products of the corresponding x and y values, ∑x is the sum of the x values, ∑y is the sum of the y values, ∑x^2 is the sum of the squares of the x values, and ∑y^2 is the sum of the squares of the y values.
By plugging in the values from the ordered pairs, we get:
n = 6
∑x = 1 + 2 + 3 + 4 + 5 + 6 = 21
∑y = 1 + 4 + 7 + 9 + 12 + 16 = 49
∑xy = (1*1) + (2*4) + (3*7) + (4*9) + (5*12) + (6*16) = 220
∑x^2 = (1^2) + (2^2) + (3^2) + (4^2) + (5^2) + (6^2) = 91
∑y^2 = (1^2) + (4^2) + (7^2) + (9^2) + (12^2) + (16^2) = 455
Now we can substitute these values into the formula:
r = (6 * 220 - 21 * 49) / sqrt((6 * 91 - 21^2) * (6 * 455 - 49^2))
= (1320 - 1029) / sqrt((546 - 441) * (2730 - 2401))
= 291 / sqrt(105 * 329)
Calculating further, we get:
r ≈ 0.8059
Therefore, the correlation coefficient that best matches the data plotted on the graph is approximately 0.8059. Since none of the answer choices exactly match this value, we can conclude that none of the given options are accurate.