STAT 516 HW 1 

HAND CALCULATIONS:

1.  We will analyze the data in Table 7.14 on pg. 341 of the textbook.  The dependent (Y) 
variable is Oxidation thickness (in micrometers) and the independent (X) variable is 
Temperature of a curing oven (in standardized units, where 0 is the 'normal' temperature), each measured on five alloys.  
Here is the data set:

Oxidation (Y): 4, 3, 3, 2, 2
Temperature (X): -2, -1, 0, 1, 2

This is a very small data set to give you a little practice in doing the linear regression calculations by hand.

Do the following by hand, SHOWING WORK.  You may use R to check your answers if you want.

(a)  Find S_yy, S_xx and S_xy.  Use these to calculate the equation of the estimated least-squares line (write the estimated regression equation)
and to find SSE and MSE.  Interpret the estimated slope and estimated intercept in the context of the variables and units in the problem.

(b)  Find a 90% confidence interval for the true slope of the regression equation.  Interpret the interval
in the context of the variables and units in the problem.

(c)  Find the residual for the first observation in the data set.

(d) Find a 95% prediction interval for the Oxidation of a new alloy having 
Temperature of 1.3.


COMPUTER CALCULATIONS:

2.  Look at the data in Table 7.19 on page 345 of the textbook.  
These data are also given in the R code link labeled 
"Heating Cost data set (plain text)" on the course web page. 

The dependent variable is kwh (electricity consumption in kilowatts per hour) and the independent 
variable is tavg (temperature in degrees) for a sample of 47 autumn days.  

Use R to answer the following questions about the data set.
Please turn in the scatter plot requested in (a) and the plots requested in (f).  
It is recommended but not required that you include in your answers the printout of the R output 
for the other parts.

(a) Does a scatter plot indicate a linear relationship between the two 
variables? Explain.

(b) Fit the least-squares regression line (using R), write the estimated regression equation,
and interpret the estimated slope in the context of this data set.  Does it make sense to 
interpret the estimated intercept?

(c) For these data, what is the unbiased estimate of the error 
variance?  (Give a number.)

(d) Using the R output, test the hypothesis that the true slope of the 
regression line is zero (as opposed to nonzero).  State the appropriate null 
and alternative hypotheses, give the value of the test statistic and give the 
appropriate P-value.  (Use a significance level of 0.05.)  Explain precisely 
what this means in terms of the relationship between the two variables.

(e)  Using R, find a 95% confidence interval for the mean electricity 
consumption for all days having average temperature 73.6 degrees.  Also, 
find a 95% prediction interval for the electricity consumption for a new 
day with average temperature 73.6 degrees.

(f)  Using R, make a residual plot and Q-Q plot of the residuals for this 
regression.  Comment about whether each of the regression assumptions seems 
to be violated, based on the plots.

(g)  Find the r^2 for this regression from the R printout, and interpret it 
in the context of this data set.


TRUE-FALSE QUESTIONS:

3.  Answer True/False Concept Questions 4,5,7,13,14,15,18,19,20 on pages 340-341.
If the statement is false, either correct it or briefly explain why it is false.