Doctor      No. of MRIs  No. of MCO Patients
1               8                        30
2              15                       10
3               4                        30
4              12                        5
5                7                       35
6              14                       10

Using the above data, create a scatterplot of the number of MRIs and the number of MCO patients.  Do the two variables seem related?
 

Part I:  Correlation Analysis

A. Compute the Pearson correlation coefficient between the MRI and MCO patient variables. What does it imply about the relationship between MRIs and MCO patients?
B. Compute the coefficient of determination (r squared).  What does it imply?
C. Test whether there is a significant negative correlation between the number of MRIs and the number of MCO patients in the population.  Show the hypotheses, sample statistic, critical statistic and decision process.  Interpret.
D.  What are the data requirements for testing hypotheses about a population correlation coefficient?
E.  What does the estimated  p value (also called the sig. level) of the test show?

F.  Excel Programming Notes:

If you know the sample size and the sample correlation coefficient, click here to open an Excel spreadsheet that will calculate the sample t value for the population correlation coefficient test. Just edit the hypothesized and sample values to fit your data set. 

G. SPSS Programming Notes:

Instead of creating the scatter plot yourself or calculating the sample correlation by hand, you can use SPSS to generate the plot and sample correlation coefficient. You must first create a data file that contains two columns of 6 numbers, the first listing the number of MRIs, and the second the number of MCO patients.   (OPTIONAL:  if you want to attach labels to the data, you can first click on the Variable View tab at the bottom of the page.  Then click on the first row’s Name box, and type in a name like mris for the first variable, and then type in an extended Label like Number of MRIs.  Then click on the second row’s Name box and type in a name like mcoers, and then an extended Label like Number of MRIs.  Then click on the Data View tab, and then type in the two columns of numbers).

(OPTIONAL) Save the data on your floppy disk/hard drive, by clicking on File, then Save As, and then the down-triangle in the Save In box.  Then click on the appropriate drive button, and then type in a name like mrimcos in the File Name box.  This will save your info on your floppy disk/hard drive in a file called mrimcos.  The next time you use SPSS you can reopen your data file by clicking on File, Open, down-triangle in the Look In box, and then on the appropriate drive button.

i.  To create a scatterplot of the MRI and MCO data, click on Graphs, then Scatter, and then Define.  Then move the MRIs and MCOs variable names into the Y-axis and X-axis boxes (it doesn’t matter which goes where).  Then click on OK.    Print out the plot by pressing on the printer icon. 

ii. To compute the sample correlation coefficients between the MRI and MCO variables, click on Analyze, then Correlate, and then Bivariate.  Then click on the MRIs and MCOs variable names and move them to the Variables box.  Then click on OK.   SPSS will then compute the Pearson correlation coefficient along with their significance levels.  Print out the results.
 

Part II:  Regression Analysis

NOTE:  Click here to access a brief guide to multiple regression analysis.

A.  Use SPSS to run a simple regression, using MRI as the dependent variable and MCO as the explanatory variable.  To do so, click on Analyze * Regression* Linear, then move the MRI variable into the Dependent variable box and the MCO variable into the Independent variable box.  Also click on the Statistics tab and click on Descriptives and Confidence Intervals and then Continue.  Then click on OK.

B.  Write the equation that describes the regression model estimated in A above. 

C.  Conduct an F test on the overall regression model.  Show the hypotheses, sample statistic, critical statistic and decision rule for the test.   What does the estimated  p value (also called the sig. level) of the test show?

D.  Test whether the number of managed care patients as any influence on the number of MRIs ordered using the t test for the population regression coefficient.  Show the hypotheses, sample statistic, critical statistic and decision rule for the test.  Interpret.

E.  What do the estimated coefficients on the explanatory variables each mean?

F.  What are the R squared and adjusted R square for the regression in D?  Interpret.

G.  Interpret the standardized coefficient.  What does it show?

H.  Interpret the 95% confidence intervals for the regression coefficients.  What do they show?

A.  Open the data file into the SPSS program by clicking on wcost99.sav.  If your computer has SPSS installed on it, the SPSS program should start and open the file.  If SPSS doesn't start right up, you can save the file to a floppy disk/local hard drive, and then start up the SPSS program, directing it to open the file at the location that you saved it to.

B.  The Wcost99.sav file contains information on the following variables for each hospital:
      i.    avgcost measures the average spending level per licensed bed at each hospital.
      ii.   beds measures the number of licensed beds at each hospital.
      iii.  staff measures the number of full-time staffers per licensed bed at each hospital.
      iv.  occup measures the occupancy rate (in percent) at each hospital.

PART I:  Correlation Analysis:

Use SPSS to compute the correlation coefficients between avgcost and beds, staff and occup.  To do so, click on Analyze * Correlate * Bivariate, then move the four variable names into the Variables box and click on OK.  Assuming Westchester hospitals are a representative sample of all US hospitals test whether average spending per bed is correlated with either hospital size, staffing or occupancy levels.  Show the hypotheses, sample statistic, critical statistic and decision rule for each test.   What do the estimated  p values (also called the sig. levels) of the tests show?

PART II:  Regression Analysis:

A.  Use SPSS to run a multiple regression, using avgcost as the dependent variable and beds, staff and occup as the explanatory variables.  To do so, click on Analyze * Regression* Linear, then move the avgcost variable into the Dependent variable box and the beds, staff and occup variables into the Independent variable box.  Also click on the Statistics tab and click on Descriptives and Confidence Intervals and then Continue.  Then click on OK.

NOTE:  Click here to access a brief guide to multiple regression analysis.

B.  Write the equation that describes the regression model estimated in A above. 

C.  Conduct an F test on the overall regression model.  Show the hypotheses, sample statistic, critical statistic and decision rule for the test.   What does the estimated  p value (also called the sig. level) of the test show?

D.  Test whether bed size, staffing levels and occupancy have any influence on average hospital spending using the t test for the population regression coefficient.  Show the hypotheses, sample statistic, critical statistic and decision rule for each test.  Interpret.

E.  What do the estimated coefficients on the explanatory variables each mean?

F.  What are the R squared and adjusted R square for the regression in D?  Interpret.

G.  Interpret the standardized coefficients.  What do they show?

H.  Interpret the 95% confidence intervals for the regression coefficients.  What do they show?