(Solved):   a. (2) Find the expected grade in the class at the end of the semester. b. (3) Assuming th ...

 

a. (2) Find the expected grade in the class at the end of the semester. b. (3) Assuming three exams are independent, find the standard deviation of "grade" at the end of the semester. c. (6) Now suppose exam 3 is independent of exam 1 and 2, but the correlation between exam 1 and 2 is \( 0.7 \). Would this change alter your answers to parts (a) and (b)? If so, how? 4. The per capita income in Cukurova region has approximately normally distributed with 15,000 TL mean and 5,000 TL standard deviation. Also, the poverty level is determined as 8,000 TL. a. (2) What proportion of population lives in poverty? b. (4) Which level of income corresponds to \( 90 \% \) of income distribution? In other words, what is the level of income that is exceeded by only \( 10 \% \) of the population? 5. (6) Indicate the lowest and highest possible values of the simple correlation coefficient \( \boldsymbol{r} \) between the variables \( X \) and \( Y \) and prove your claim. 6. Consider sample size of \( T \) on an independently and identically distributed random variable \( Y \) with mean \( \mu_{y} \) and variance \( \sigma_{y}^{2} \). Suppose you calculate the variance for \( Y \) as \( s^{2}=\left[\frac{\sum_{t=1}^{T}\left(y_{t}-\bar{y}\right)^{2}}{T}\right] \) a. (3) Show that \( \operatorname{var}(\bar{y})=\sigma_{y}^{2} / T \). b. (4) Find \( E\left[s^{2}\right] \). c. (3) Is \( E\left[s^{2}\right]=\sigma_{y}^{2} \) ? If not, what modification would you need in the formula for \( s^{2} \) ? d. (3) Interpret your answers in (b) and (c) using the concept of "degrees of freedom." 7. Take the simple regression model, \( y_{t}=\beta_{1}+\beta_{2} x_{t}+e_{t} \), and consider that all usual assumptions, including normality are satisfied. a. (3) Find the expected value of the least squares residual \( \hat{e}_{t} \). Show your work. b. (4) Show that \( E(\bar{y})=E\left[\frac{\sum_{t=1}^{T} y_{t}}{T}\right]=\beta_{1}+\beta_{2} \bar{x} \), where \( \bar{x}=\sum_{t=1}^{T} x_{t} / T \). Clearly state any assumptions you are employing at each step of your answer. c. (3) Show that \( \operatorname{var}(\bar{y})=\sigma^{2} / T \). Clearly state any assumptions you are employing at each step of your answer. d. (5) "If I am running a regression of \( y \) on \( x \), it is better to have all the \( x \) values bunched up and close together, so that I have a better idea of where the least squares regression line goes." Is this statement true or false? Explain your answer.