3 Using the properties of the expected value operator prove

3. Using the properties of the expected value operator, prove that the population variance EL(X - E(X))2] is equivalent to the expression E(X2) -[E(X)]2. 4. Consider the following data gathered from students concerning how many hours per week they study, and their score on the resulting quiz Student Hours Studied Test Score James Cara Elliot Erin 40% 60% 0% 90% (a) Suppose I want to use this data to get information about the effect of studying on quiz scores. Write out the correct simple linear regression model for this question (b) Calculate the parameter estimates for the intercept and slope, B1 and B2 (c) Interpret B1, i.e. what does this parameter estimate tell us?. (d) Interpret B2, i.e. what does this parameter estimate tell us? e) Calculate the coefficient of determination, R2 for this regression. What does this tell us about the relationship between hours studied and test scores? (f) What is the predicted test score for a student who studies for 2.5 hours? (g) Suppose a student wanted to score 100% on the quiz. How many hours would they have to study to expect a score of 100%?

Solution

ANS :

TO PROVE: Var(X) = E[ X - (E(X)) ] ² = E(X) - [E(X)]²  

PROOF:

Var(X)=E[(X?E[X])2]

Var(X)=E[(X?E[X]) (X?E[X])]

Var(X)=E[X² ?2XE[X]+ (E[X])² ]

Var(X)=E[X²] ? 2E[X] E[X] +(E[X])² as E (E[X])² = (E[X])² ; E[XE[X]] = E[X] E[X]

Var(X)=E[X²] - 2 (E[X])² + (E[X])²

Var(X)=E[X²] - (E[X])²

hence, proved.