Exercise 1:

A new profit-sharing plan was introduced at an automobile parts manufacturing plant last year. Both management and union representatives were interested in determining how a worker's years of experience influence his or her productivity gains. After the plan had been in effect for a while, the data shown below were collected:

Years of experience     Number of units

_i=1∑⁹ y_i = 965, _i=1∑⁹ x_i = 133.9, _i=1∑⁹ y²_i = 104469, _i=1∑⁹ x²_i = 2258.73, _i=1∑⁹ x_iy_i = 1480.12

a. Find the least-squares regression line (perform all calculations by hand).

b. Use R: Verify your answer to part (b). Submit the R results.

c. Use R: Construct the scatterplot of the number of units manufactured daily against the number of years of experience and add the fitted line .

d. Predict the number of units manufactured daily by an employee who has 10 years of experience on the assembly line.

Exercise 2

For the regression model y_i = β₀ + β₁x_i + ∈_i show that

a. ∑ⁿ_i=1 e_i = 0,

b. ∑ⁿ_i=1 e_ix_i = 0,

c. ∑ⁿ_i=1 e_iy^{^}_i = 0,

Exercise 3:

Suppose in the model y_i = β₀ + β₁x_i + ∈_i, where i = 1, ........, n, E(∈_i) = 0, var(∈_i) = σ² the measurements x_i were in inches and we would like to write the model in centimeters, say, z_i. If one inch is equal to c centimeters (c is known), we can write the above model as follows y_i = β₀^*+ β₁^*z_i + ∈_i.

a. Suppose β*₀ and β^{^}₁ are the least squares estimates of β₀ and β₁ of the first model. Find the estimates of β₀ and β^* in terms of β^*₀ and β^*₁.

b. Show that the value of R² remains the same for both models.

c. Find the variance of β^*₁^{^}

Exercise 4:

Consider the regression model

y_i = (β₀ + β₁x¯) + β₁(x_i - x¯) + ∈_i

This model is called the centered version of the regression model y_i = β₀ + β₁x_i + ∈_i that was discussed in class. If we let γ₀ = β₀ + β₁x¯ we can rewrite the centered version as y_i = γ₀ + β₁(x_i - x¯) + si. Find the least squares estimates of γ₀ and β₁.

Exercise 5:

Consider the regression model y_i = β₀ + β₁x_i + ∈_i. Show that Cov(Y¯ , β^*₁) = 0 where Y¯ is the sample mean of the y values, and β^*₁ is the estimate of β₁.

Exercise 6:

Consider the regression model y_i = β₀ + β₁x_i + ∈_i. Find cov(e_i, e_j ).

Exercise 7:

Suppose Y_i = βx_i + ∈_i. In this equation X is non-random, β is a parameter (unknown), and ∈ ~ N (0, σ).

a. Find the mean of Y .

b. Find the variance of Y .

c. What distribution does Y follow?

d. Write down the likelihood function based on n observations of Y and x.

e. Find the maximum likelihood estimate of β. Denote it with β^{^}.

f. Show that the estimate of part (e) is unbiased estimator of β.

g. Find the variance of this estimate.

Exercise 8:

Find the covariance between e_i and e_j . Also find the variance of e_i.