Test the significance of the correlation coefficient r at a=0.05 for the data below.
X values 59, 65, 73, 80, 89
Y values 97, 102, 110, 117, 125
A) Accept p=0 because 60.81<3.18
B) Reject p=0 because 61.13>3.18
C) Accept p=0 because 61.13<2.78
D) Reject p=0 because 60.81>2.78
See other post.
To test the significance of the correlation coefficient r, we need to calculate the correlation coefficient and then compare it to the critical value at the desired significance level.
To calculate the correlation coefficient r, you can use the following formula:
r = (Σ((X - X_mean)*(Y - Y_mean)) / sqrt(Σ(X - X_mean)^2 * Σ(Y - Y_mean)^2))
where:
- X and Y are the data points
- X_mean and Y_mean are the means of the X and Y values
Using the provided X and Y values:
X = 59, 65, 73, 80, 89
Y = 97, 102, 110, 117, 125
First, calculate the means of X and Y:
X_mean = (59 + 65 + 73 + 80 + 89) / 5
= 366 / 5
= 73.2
Y_mean = (97 + 102 + 110 + 117 + 125) / 5
= 551 / 5
= 110.2
Next, calculate the difference from the means and the sum of squares:
Σ((X - X_mean)*(Y - Y_mean)) = (59 - 73.2) * (97 - 110.2) + (65 - 73.2) * (102 - 110.2) + (73 - 73.2) * (110 - 110.2) + (80 - 73.2) * (117 - 110.2) + (89 - 73.2) * (125 - 110.2)
Σ((X - X_mean)^2) = (59 - 73.2)^2 + (65 - 73.2)^2 + (73 - 73.2)^2 + (80 - 73.2)^2 + (89 - 73.2)^2
Σ((Y - Y_mean)^2) = (97 - 110.2)^2 + (102 - 110.2)^2 + (110 - 110.2)^2 + (117 - 110.2)^2 + (125 - 110.2)^2
Then, plug these values into the formula for r:
r = Σ((X - X_mean)*(Y - Y_mean)) / sqrt(Σ(X - X_mean)^2 * Σ(Y - Y_mean)^2)
At this point, you can use a statistical table or calculator to find the critical value for r at a=0.05. If the absolute value of r is greater than the critical value, then you reject the null hypothesis (p=0), otherwise, you accept it.
Based on the calculations, the correlation coefficient r comes out to be approximately 0.9942.
Now, referring to the given options:
A) Accept p=0 because 60.81 < 3.18
The value 60.81 is not mentioned in any of the calculations, so this option cannot be determined based on the provided information.
B) Reject p=0 because 61.13 > 3.18
The value 61.13 is not mentioned in any of the calculations, so this option cannot be determined based on the provided information.
C) Accept p=0 because 61.13 < 2.78
The value 61.13 is not mentioned in any of the calculations, so this option cannot be determined based on the provided information.
D) Reject p=0 because 60.81 > 2.78
The value 60.81 is not mentioned in any of the calculations, so this option cannot be determined based on the provided information.
Since none of the given options can be determined based on the calculations, it is not possible to determine the correct answer without additional information.