Roll a fair, six-sided die 28 times. Let X be the sum of the first 10 rolls. Let Y be the sum of the last 10 rolls. Find the correlation between X and Y
To find the correlation between X and Y, we need to calculate the covariance and standard deviations of both X and Y.
But before we do that, let's find the mean and standard deviation of a single roll of the six-sided die.
Mean of a single roll = (1+2+3+4+5+6)/6 = 3.5
Next, we'll calculate the standard deviation. To do this, we'll need to calculate the variance first.
Variance = (1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2)/6 - (mean)^2
= (1+4+9+16+25+36)/6 - (3.5)^2
= 91/6 - 12.25
= 15.1667
Standard Deviation = √Variance = √15.1667 = 3.891
Now that we have the mean and standard deviation of a single roll, we can calculate the mean and standard deviation of X and Y.
Mean of X = mean of a single roll * number of rolls = 3.5 * 10 = 35
Standard Deviation of X = standard deviation of a single roll * √number of rolls = 3.891 * √10
Mean of Y = mean of a single roll * number of rolls = 3.5 * 10 = 35
Standard Deviation of Y = standard deviation of a single roll * √number of rolls = 3.891 * √10
Next, we'll calculate the covariance of X and Y using the formula:
Covariance(X,Y) = (sum of (Xi - mean of X)(Yi - mean of Y))/(number of rolls - 1)
Since we rolled the die 28 times, the covariance formula becomes:
Covariance(X,Y) = (sum of (Xi - 35)(Yi - 35))/(28 - 1)
Finally, we can calculate the correlation coefficient using the formula:
Correlation coefficient = Covariance(X,Y) / (Standard Deviation of X * Standard Deviation of Y)
By substituting the values we calculated above, we can find the correlation between X and Y.