Students took two parts of a test, each worth 50 points. Part A has a variance of 25, and Part B has a variance of 49. The correlation between the test scores is 0.6.

(a) If the teacher adds the grades of the two parts together to form a final test grade, what would the variance of the final test grades be?

(b) What would the variance of Part A - Part B be?

## Solution

Given that `$V(\text{ Part A})= 25$`

and `$V(\text{Part B})= 49$`

. Also the correlation between the test scores `$r = 0.6$`

.

` $$ \begin{aligned} Cov(\text{Part A},\text{Part B}) &=r\sqrt{V(\text{Part A})V(\text{Part B})} \\ &=0.6\times \sqrt{25*49}\\ &=21. \end{aligned} $$ `

(a) Suppose the teacher adds the grades of the two parts together to form a final test grade.

That is `$\text{Final grade} = \text{Grade of Part A} + \text{Grade of Part B}$`

.

Then the variance of Final grade is

` $$ \begin{aligned} V(\text{Final grade}) &= V(\text{Part A} + \text{Part B})\\ &= V(\text{Part A}) + V(\text{Part B})+ 2 Cov(\text{Part A}, \text{Part B})\\ &= 25 + 49 +2*21\\ &= 116 \end{aligned} $$ `

(b) The variance of `$\text{Part A} - \text{Part B}$`

is

` $$ \begin{aligned} V(\text{Part A} - \text{Part B})&= V(\text{Part A}) + V(\text{Part B})- 2 Cov(\text{Part A}, \text{Part B})\\ &= 25 + 49 -2*21\\ &= 32 \end{aligned} $$ `

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators