x-value: 1,5,7,11,15,17.
y-value: 23,27,32,35,39,43.
what is the r-value to two decimal places, if necessary
To calculate the r-value (correlation coefficient) between the x-values and y-values, you can use statistical software or a calculator that has this function.
Using statistical software, you would input the x-values and y-values, and it would provide you with the r-value.
If you want to calculate it manually, you can follow these steps:
Step 1: Calculate the mean of both x-values and y-values:
mean of x-values (x̄) = (1+5+7+11+15+17)/6 = 9.33
mean of y-values (ȳ) = (23+27+32+35+39+43)/6 = 32.33
Step 2: Calculate the sum of the products of the differences between each x and y value and the mean of each:
Σ ((x_i - x̄) * (y_i - ȳ))
For our data:
[(1-9.33)*(23-32.33)] + [(5-9.33)*(27-32.33)] + [(7-9.33)*(32-32.33)] + [(11-9.33)*(35-32.33)] + [(15-9.33)*(39-32.33)] + [(17-9.33)*(43-32.33)]
= (-8.33)*(-9.33) + (-4.33)*(-5.33) + (-2.33)*(-0.33) + 1.67*2.67 + 5.67*6.67 + 7.67*10.67
= 77.7789 + 23.1129 + 0.7669 + 4.459 + 37.8389 + 81.7029
= 225.6586
Step 3: Calculate the sum of the squared differences for x-values and y-values:
Σ (x_i - x̄)^2 = (1-9.33)^2 + (5-9.33)^2 + (7-9.33)^2 + (11-9.33)^2 + (15-9.33)^2 + (17-9.33)^2
= 68.8689 + 18.1089 + 7.1089 + 2.7189 + 29.2389 + 63.1489
= 189.1824
Σ (y_i - ȳ)^2 = (23-32.33)^2 + (27-32.33)^2 + (32-32.33)^2 + (35-32.33)^2 + (39-32.33)^2 + (43-32.33)^2
= 86.2689 + 28.0129 + 0.1089 + 7.4389 + 45.2289 + 121.8089
= 288.8674
Step 4: Calculate the correlation coefficient (r):
r = Σ ((x_i - x̄) * (y_i - ȳ)) / √(Σ (x_i - x̄)^2 * Σ (y_i - ȳ)^2)
r = 225.6586 / √(189.1824 * 288.8674)
r = 225.6586 / √(54670.8414)
r = 225.6586 / 233.8285
r = 0.964
Therefore, the correlation coefficient (r-value) between the x-values and y-values is approximately 0.96.