Using a scatter diagram with x(age in weeks) of calves and y (weight in kg) and the info is as follows
Week 1 /42kgs
Week 3 /50 kgs
Week 10/75 kgs
Week 16/100kgs
Week 26/150kgs
week 36/200kgs
would you estimate the linear correlation coefficient R to be positive , close to zero, or negative?
I say it would be positive but I am then unsure what formula is how I compute the sample correlation coefficient r?
To estimate the linear correlation coefficient (R), we need to calculate the sample correlation coefficient (r). The formula to compute the sample correlation coefficient is as follows:
r = (nΣXY - ΣXΣY) / √[(nΣX² - (ΣX)²)(nΣY² - (ΣY)²)]
Here's how you can compute the sample correlation coefficient (r) for the given data:
Step 1: Calculate the values nΣXY, ΣX, ΣY, nΣX², and nΣY², where:
- n represents the number of data points (in this case, it is 6)
- ΣXY represents the sum of the products of X and Y values (age in weeks and weight in kg)
- ΣX represents the sum of the X values (age in weeks)
- ΣY represents the sum of the Y values (weight in kg)
- ΣX² represents the sum of the squares of the X values
- ΣY² represents the sum of the squares of the Y values
Week 1: X = 1, Y = 42
Week 3: X = 3, Y = 50
Week 10: X = 10, Y = 75
Week 16: X = 16, Y = 100
Week 26: X = 26, Y = 150
Week 36: X = 36, Y = 200
nΣXY = (1*42) + (3*50) + (10*75) + (16*100) + (26*150) + (36*200)
ΣX = 1 + 3 + 10 + 16 + 26 + 36
ΣY = 42 + 50 + 75 + 100 + 150 + 200
nΣX² = (1²) + (3²) + (10²) + (16²) + (26²) + (36²)
nΣY² = (42²) + (50²) + (75²) + (100²) + (150²) + (200²)
Step 2: Substitute the values into the formula and calculate the numerator and denominator.
Step 3: Divide the numerator by the denominator to obtain the sample correlation coefficient (r).
The resulting value of r will indicate the strength and direction of the linear relationship between age (X) and weight (Y). If r is positive, it indicates a positive linear relationship. If r is close to zero, it indicates a weak or no linear relationship. If r is negative, it indicates a negative linear relationship.
Calculation of r will give you the estimate of the linear correlation coefficient (R) for the given data.