The purity of oxygen produced by a fractional distillation process is thought to be related to the percentage of hydrocarbons in the main condensor of the processing unit

Homework 1:

This homework will be due on Sep 30, 2021

1. The purity of oxygen produced by a fractional distillation process is thought to be related to the percentage of hydrocarbons in the main condensor of the processing unit. Twenty samples are shown below.
2. Fit a simple linear regression model to the data, find the fitted line and calculate .
3. Test the hypothesis H 0 : β 1 = 0.
4. Find a 95% CI on the slope.
5. Find a 95% CI on the mean purity when the hydrocarbon percentage is 1.00.
6. Find a 95% prediction interval for purity when the hydrocarbon percentage is 1.00.

1. On March 1, 1984, the Wall Street Journal published a survey of television advertisements conducted by Video Board Tests, Inc., a New York ad – testing company that interviewed 4000 adults. These people were regular product users who were asked to cite a commercial they had seen for that product category in the past week. In this case, the response is the number of millions of retained impressions per week. The regressor is the amount of money spent by the firm on advertising. The data follow.

1. Fit the simple linear regression model to these data, and find the fitted line.
2. Is there a significant relationship between the amount a company spends on advertising and retained impressions? Use R outcome and hypothesis test to justify your answer.
3. Give the 90% confidence interval for the mean number of retained impressions for MCI given the amount of money spent by the firm is 195.
4. Give the 90% prediction interval for the number of retained impressions for MCI given the amount of money spent by the firm is 195.

1. Consider the simple linear regression model , with E ( ε ) = 0, Var( ε ) = , and ε uncorrelated.

1. Show that Cov( ) = .

1. b. Write down the Least square estimators (LSE) and maximum likelihood estimators (MLE) of and .

1. c. Find the expectation and variance of LSE of

1. Bonus (0.5 pt)

Consider the simple linear regression model , with E ( ε ) = 0, Var( ε ) =  , and ε uncorrelated.

1. Show that E( ) =

where