Questions: The following is a set of data from a sample of n=11 items. Complete parts (a) through (c).
X 20 6 11 14 13 17 16 12 10 1 7
Y 40 12 22 28 26 34 32 24 20 2 14
a. Compute the sample covariance.
(Round to three decimal places as needed.)
Transcript text: The following is a set of data from a sample of $n=11$ items. Complete parts (a) through (c).
\[
\begin{array}{rrrrrrrrrrrr}
\mathbf{X} & 20 & 6 & 11 & 14 & 13 & 17 & 16 & 12 & 10 & 1 & 7 \\
\mathbf{Y} & 40 & 12 & 22 & 28 & 26 & 34 & 32 & 24 & 20 & 2 & 14
\end{array}
\]
a. Compute the sample covariance.
$\square$ (Round to three decimal places as needed.)
Solution
Solution Steps
Step 1: Calculate the Sample Covariance
The sample covariance between the variables X and Y is calculated using the formula:
Cov(X,Y)=n−11i=1∑n(Xi−Xˉ)(Yi−Yˉ)
For the given data, the calculated sample covariance is:
Cov(X,Y)=58.945
Step 2: Calculate the Standard Deviations
The standard deviation for X and Y is calculated using the formulas:
σX=n−11i=1∑n(Xi−Xˉ)2
σY=n−11i=1∑n(Yi−Yˉ)2
The calculated standard deviations are:
σX=5.429
σY=10.858
Step 3: Calculate the Correlation Coefficient
The correlation coefficient r is calculated using the formula:
r=σXσYCov(X,Y)
Substituting the values:
r=5.429×10.85858.945=1.0
Final Answer
The sample covariance is 58.945 and the correlation coefficient is 1.0.