Find the least-squares regression line - find the least

Question

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

Years of	Number of units
experience (x)	daily (y)
15.1	110
7.0	105
18.6	115
23.7	127
11.5	98
16.4	103
6.3	87
15.4	108
19.9	112

For your convenience:

_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.