Lab for the week of 03APR

Getting the data into SAS

As usual, we need to submit some SAS code from the program editor in order to read in the dataset. We will be reading the data from a text file in my home directory using the code

libname sta110b '~/sta110b';
data sta110b.bigbang;
  infile '~jls11/public/bigbang.dat';
  input velocity distance;
run;

Click on the "running man" icon to submit the code. To start INSIGHT, type "insight" on the command line of the toolbox window. Choose your data set from among those listed to begin the analysis.

About the data

These observations of nebulae outside the Milky Way were made by Edwin Hubble using the Mount Wilson Observatory. He found a relationship between a nebula's distance from earth and the velocity at which it was moving away from Earth. The distances in this dataset are measured in megaparsecs and the velocities in kilometers per second. As you can gather, the measurement techniques for distances were not extremely accurate. Hubble's findings (published as "A Relation Between Distance and Radial Velocity Among Extra-galactic Nebulae", Proceedings of the National Academy of Science 15 (1929): 168-73) led to the formulation of theories as to why distance and velocity seemed to be related; eventually, the Big Bang Theory emerged as the dominant theory. Basically, the idea is that the universe began a long, long time ago with a "big bang". The material in the universe traveled out a constant velocity from this point, which helps explain the relationship Hubble observed.

Questions

Please note that this lab doesn't include the same level of specificity in the directions as some of the previous labs. This may require some additional thought, but use what you learned in previous labs and don't be afraid to explore the options available in SAS:INSIGHT. It's OK to make some mistakes; the labs aren't graded so that you can feel free to learn by trial and error. This will help you later with your project.

1)Draw the scatter plot (under the Analyze menu) with distance on the y-axis and velocity on the x-axis. Does it seem reasonable to fit a linear regression to this data?

2)Try fitting a linear regression to the data (using Fit under the Analyze menu). The resulting output contains a information that will help you in consideration of the following questions.

3)What are the estimates for the slope and intercept of the fitted line?

4)Find the test statistic and p-value for test whether there is a relationship between distance and velocity.

5)Does it appear the residuals are independent and normally distributed with mean 0 and some variance? (Remember that this is one of the assumptions of our method.) To obtain the residuals as an additional column in the data set, choose "Residual" under Vars. Then you draw a histogram and a boxplot of the residuals (using Distribution under Analyze) to get an idea about their distribution. Another interesting plot (which will we will be the subject of further discussion), is the plot of residuals vs. fitted values (included at the bottom of the regression output). You should the points scattered without any apparent pattern and with fairly even spread around a mean of 0.

6)What is the 95% confidence interval for the beta, the slope of the regression line? How can you calculate 90% or 99% confidence intervals? (Hint: Try the Tables menu from within the regression output window. For simple linear regression, both "LR" and "Wald" options should give the same output.)

7)Re-fit the regression model, but this time select the particular outputs you want (confidence intervals, residual plots, etc.) before you fit the model. To do this, use Fit under Analyze as before, but when you enter the window in which you select the variables, click on the Output button to make your choices before allowing the fitting of the model to proceed.