MTH205 Introduction to Statistical Methods

Tutorial 6

Based on Chapter 6

1. For a simple linear regression model

Yi = β0 + β1xi + ; i = 1, 2,...,n

where the are independent and normally distributed with zeros means and equal variances σ2, the estimators for β1 and β0 are respectively

Denoting show that

2. For a simple linear regression model

Yi = β0 + β1xi + ; i = 1, 2,...,n

where the are independent and normally distributed with zero means and equal variances σ2, show that ¯Y and the estimator

are uncorrelated.

3. The 18 measurements of the amounts y of a chemical compound that dissolved in 100 grams of water at various temperatures x were recorded as follows:

The data is fitted using a simple linear regression model

Yi = β0 + β1xi + ; i = 1, 2,..., 18

where ’s are independent and normally distributed with zero means and equal variances σ2.

(i) Obtain the least squares estimates of β0 and β1.

(ii) Estimate the amount of chemical that will dissolve in 100 grams of water at 50◦C.

(iii) Evaluate s2, the unbiased estimate of σ2.

(iv) Construct a 99% confidence interval for β0.

(v) Construct a 99% confidence interval for β1.

(vi) Does the temperature have a significant influence on the amounts y of chemical compound that can dissolve in 100 grams of water? Use a p-value in your conclusion.

(vii) Construct a 99% confidence interval for the average amount of chemical that will dissolve in 100 grams of water at 50◦C.

(viii) Construct a 99% prediction interval for the amount of chemical that will dissolve in 100 grams of water at 50◦C.

4. For the simple linear regression model, show that

That is, show that

Total variation = Explained variation + Unexplained variation.

5. The following data is given:

x   0 1 2 3 4 5 6

y   1 4 5 3 2 3 4

(i) Fit the cubic regression equation yˆ = βˆ0 + βˆ1x + βˆ2x2 + βˆ3x3.

(ii) Predict yˆ when x = 2.