The following table gives the average annual tuition (y, in dollars) at a privat

Question

The following table gives the average annual tuition (y, in dollars) at a private college for the years 1995-2009, where x = 1 corresponds to the year 1995. Suppose a simple linear regression model Y = beta_0 + beta_1 x + epsilon with epsilon ~ N(0, sigm^2) is used to describe the data. (a) Find least-squares estimators for beta_0 and beta_1 and an unbiased estimator for sigma^2. (b) Find the value of the coefficient of determination r^2 and draw the residual plot for the data. We want to fit an exponential model Y = alpha_0(alpha_1)^x for the data in problem Note that taking the natural logarithm leads to the linear model ln Y = beta_0 + beta_1 x + epsilon_1 where beta_0 = ln alpha_0, beta_1 = ln alpha_1, and epsilon ~ N(0, sigma^2) (a) Find least-squares estimators for beta_0 and beta_1 and an unbiased estimator for sigma^2. (b) Find the value of the coefficient of determination r^2 and draw the residual plot for the data.

Explanation / Answer

lnY = B0 + B1*x + e

Z = B0 +B1*x

Summation on both sides

Z = B0 n + B1 * x

multiplying with x

Z*x = B0 * x+ B1 * x^2

Substituting above values

B0 = 9.104241

B1 = 0.067056

A0 = 8993.352

A1=1.069355

b)

x y lny=Z x^2 X*Z 1 9933.33 9.203651 1 9.203651 2 9450.2 9.153791 4 18.30758 3 10715.7 9.279465 9 27.8384 4 11861.1 9.381019 16 37.52408 5 13317.6 9.496842 25 47.48421 6 13428 9.505097 36 57.03058 7 15512.1 9.649376 49 67.54563 8 15229 9.630957 64 77.04765 9 16582.4 9.716097 81 87.44487 10 15271.6 9.63375 100 96.3375 11 19884.6 9.897701 121 108.8747 12 21719 9.985943 144 119.8313 13 20437.2 9.925112 169 129.0265 14 24599.2 10.11047 196 141.5466 15 22948.2 10.04099 225 150.6149 SUM 120 144.6103 1240 1175.658

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