MTH205 Introduction to Statistical Methods Tutorial 6Java

Java Python 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 s MTH205 Introduction to Statistical Methods Tutorial 6Java quares 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