Data covering the most recent $30$ days are given in the following table for the price per gallon of regular gasoline at a local station.

Day Price Day Price

$12\mathrm{.}53162\mathrm{.}46$

$22\mathrm{.}35172\mathrm{.}60$

$31\mathrm{.}91182\mathrm{.}10$

$42\mathrm{.}20192\mathrm{.}01$

$51\mathrm{.}77202\mathrm{.}14$

$63\mathrm{.}26212\mathrm{.}03$

$71\mathrm{.}63222\mathrm{.}68$

$82\mathrm{.}73232\mathrm{.}59$

$92\mathrm{.}41242\mathrm{.}99$

$102\mathrm{.}72252\mathrm{.}94$

$112\mathrm{.}87261\mathrm{.}77$

$121\mathrm{.}49272\mathrm{.}62$

$132\mathrm{.}92283\mathrm{.}19$

$143\mathrm{.}53293\mathrm{.}01$

$152\mathrm{.}74302\mathrm{.}10$

a$.$Compute a forecast for the next day's price by using a $3-$day moving average. What is the forecast? What is the MSE for the last five observations?

b$.$Can you reduce the MSE for the last five observations by changing the moving$-$average window? $($Try $4-$ to $10-$day windows.$)$

$$c$.$ Compute a forecast for the next day's price by using exponential smoothing with a $=0\mathrm{.}25.$ What is the forecast? What is the MSE for the last five

observations?

d$.$Can you improve the MSE for the last five observations by changing the smoothing constant a$?$