X 3, 5, 6, 8, 9, 10, 12, 14, 15
Y 30,33,34,36,37,40, 50, 60, 70
Calculate Linear Correlation Coefficient r
Use significance level 0.05 and Appendix Table A-6 to make decision
about linear correlation between x and y.
r = (Σxy)/N(SDx)(SDy)
Calculate each term and solve.
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To calculate the linear correlation coefficient (r) between two sets of data, we need to follow these steps:
Step 1: Calculate the means of both sets of data (x and y).
The mean of x (mean_x) can be calculated by adding up all the values of x and dividing by the number of data points:
mean_x = (3 + 5 + 6 + 8 + 9 + 10 + 12 + 14 + 15) / 9
The mean of y (mean_y) can be calculated in the same way:
mean_y = (30 + 33 + 34 + 36 + 37 + 40 + 50 + 60 + 70) / 9
Step 2: Calculate the deviation of each data point from the mean.
To calculate the deviation of each x value from the mean_x, subtract the mean_x from each x value:
deviation_x = (3 - mean_x), (5 - mean_x), (6 - mean_x), (8 - mean_x), (9 - mean_x),
(10 - mean_x), (12 - mean_x), (14 - mean_x), (15 - mean_x)
Do the same for the y values to get the deviation_y.
Step 3: Calculate the product of the deviations.
Multiply each deviation_x with the corresponding deviation_y to get the product of the deviations for each data point.
Step 4: Sum up the products of the deviations.
Sum up all the products of the deviations calculated in the previous step.
Step 5: Calculate the standard deviation of x and y.
To calculate the standard deviation of x (std_dev_x), we need to square each deviation_x, calculate the mean of these squared values, and then take the square root of the mean. The same process applies to calculate the standard deviation of y (std_dev_y).
Step 6: Calculate the correlation coefficient (r).
The correlation coefficient (r) is calculated by dividing the sum of the products of the deviations by the product of the standard deviations of x and y.
r = (sum of products of the deviations) / (std_dev_x * std_dev_y)
Now, to make a decision about the linear correlation between x and y using a significance level of 0.05 and Appendix Table A-6 (which contains critical values of r for different sample sizes), we need to compare the calculated r value with the critical value of r from the table.
Find the row in the table corresponding to the sample size (in this case, the sample size is 9) and find the column corresponding to the significance level (0.05). This intersection will give you the critical value of r.
If the calculated r value is greater than the critical value of r, there is a significant linear correlation between x and y. If the calculated r value is less than the critical value of r, there is no significant linear correlation between x and y.