Answer the following questions about correlation (r)

1. (5 points)

Facts about correlation. 

Answer the following questions about correlation (r).

a.       What is the strongest the correlation can ever be?

b.      If there is no relationship, r is equal to

c.       The correlation coefficient ranges from __________to __________

d.      If the points fall in an almost perfect, negative linear pattern, r is close to: __________

e.       If the points fall in an almost perfect, positive linear pattern, r is close to:  __________

2.  (6 points)

Relationship between Height and Weight.

Data has been collected on 219 STAT 200 students. Weight is measured in pound and Height in inch. Below are some descriptive statistics of Weight and Height.

Then a linear regression was performed on height and weight. The output looks as follows:

a.       Write the regression equation based on the output.

b.      What is the response variable (dependent variable) and what is the predictor (independent variable)?

c.    Based on the equation, what is the slope? Please explain slope as the change in Y per unit change in X in the context of the variables used in this problem.

d.    Based on the output, what is the test of the slope for this regression equation?  That is, provide the null and alternative hypotheses, the test statistic, p-value of the test, and state your decision and conclusion.

e.       Assume a student is 65 inch tall.  Is it possible to predict his weight based on this analysis? If so, please estimate his weight using the regression equation.

f.     What do the Fitted (predicted) values and Residuals represent? For example, there is one record in the data set with height = 54 and weight = 110. Please use these numbers to explain what is the fitted value and what is the residual.

3. (11 points) Parts B, D are worth 2 points.  Parts C, E are worth 3 points (2 plots and a conclusion for each)

Relationship between eighth grade IQ and ninth grade math score.

For a statistics class project, students examined the relationship between x = 8^th grade IQ and y = 9^th grade math scores for 20 students.  The data are displayed below.

a.       Create a scatter plot of the measurements by selecting Math Score for the y-axis (response) and IQ for the x-axis (predictor).  Describe the relationship between math score and IQ.

Minitab Users: Graph > Scatter Plot > Simple.

SPSS Users: Graphs > Legacy Dialogues > Scatter/Dot > Simple Scatter

b.      Perform a linear regression with the Response (dependent variable) math score and the variable IQ as the Predictor (independent variable).  Store/Save the (unstandardized)  Residuals and Fitted(Predicted) values. These will be stored in the fourth and fifth columns of the data worksheet.

What is the regression equation?

What is the interpretation of R-square (just use the latest output) and how to calculate correlation based on it?

c.       One of the students with a high IQ (number 17) appears to be an outlier.  With a sample size of only 20 this can affect our normality assumption.  Also, the constant variance assumption could be compromised.  We can visually check for constant variance using a Residual Plot and test for normality using a Probability Plot (or Q-Q plot).

To get a residual plot, simply create a Scatterplot using the Residuals  as the y-variable and the Fitted(Predicted) Values as the x-variable. (Remember these should have been stored/saved when you first performed the regression per instructions above.  If not, re-run regression and click store/save and click the boxes for unstandardized residuals and fits(predicted) values.)  Now create a probability plot (Q-Q plot if using SPSS) of the residuals. 

Based on these two graphs and what you have learned about hypothesis testing, what interpretations do you come to regarding the assumptions of constant variance and normality?

Minitab Users: Probability plot go to Graphs > Probability Plot > Single and select Residuals

SPSS Users: Q-Q plot with normal test go to Analyze > Descriptive Statistics > Explore and enter Unstandardized Residuals in Dependent List click Plots and select box for Normal plots with tests

d.    Although outliers should never be deleted without a reason, there are several reasons why it may be legitimate to conduct an analysis without them. Delete the data point for row 17 (click on the cell with the IQ of 114, enter * and then click on any other cell - this “enters” the asterisk in that previous cell. ) and re-calculate the regression line for the remainder of the data.

What is the regression equation with the rest of the data?

What is the R² and correlation between Math Score and IQ with the outlier removed?

e.    How does the fit of the regression line of the original data (i.e. with outlier) compare (visually and statistically) to the fit of the regression line to the data with the outlier removed?  Compare the fit of the regression line between the two sets of data.  Pay particular attention to the differences in R², the slope and how the line fits each set of data.  Repeat the residual plot and probability plot!