1. Facts About Correlation, r.

1. Facts About Correlation, r.Answer the following questions about the correlation coefficient, r.a. The correlation coefficient ranges from ____ to ____.b. If there is no relationship, r is equal to ____.c. If the points fall in an almost perfect, negative linear pattern, r is close to: ____.d. If the points fall in an almost perfect, positive linear pattern, r is close to: ____.2. Linear Regression – Interpreting Software OutputRelationship between Height and Weight: Height and weight data has been collected on 219 STAT 200 students. The variable “Weight” was measured in pounds and “Height” was measured in inches. Below are the descriptive statistics.Since both height and weight are quantitative, linear regression was performed. We want to see if height is a significant linear predictor of weight. The Minitab (left) and SPSS (right) output is below:a. Using the output, write the regression equation.b. What is the response variable (dependent variable)?c. What is the predictor variable (independent variable)?d. Based on the regression equation, what is the value of the slope?e. Interpret the slope in the context of the problem.Hint: In general, the slope is interpreted as the change in Y per unit change in X.f. 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.i. Null hypothesis:ii. Alternative hypothesis:iii. Test statistic:iv. P-value:v. Statistical conclusion (reject or fail to reject the null):vi. Real world conclusion:g. Assume that a student is 65 inches tall. Estimate his or her weight using the regression equation. First, consider if this would be a reasonable value to estimate from the equation.h. Explain Fitted (predicted) values and Residuals. Calculate the fit and residual for the row in the data set where height = 54 and weight = 110.i. Explain fitted values:ii. Explain residuals:iii. Find Fit for height = 54 and weight = 110:iv: Find Residual for height = 54 and weight = 110:Hint: Remember that residual = observed response – fitted value.3. Linear Regression – Using SoftwareRelationship between eighth grade IQ and ninth grade math score: Open the IQ data set. For a statistics class project, students examined the relationship between x = 8th grade IQ and y = 9th grade math scores for 20 students.a. Create a scatter plot of the data using Math Score as the response (y-axis) and IQ as the predictor (x-axis). Copy and paste the scatter plot. Use the scatter plot to describe the relationship between the variables.• Minitab Users: Graph > Scatter Plot > Simple.• SPSS Users: Graphs > Legacy Dialogues > Scatter/Dot > Simple Scatterb. Perform a linear regression using Math Score as the response variable (dependent variable) and 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.Copy and paste your output.c. What is the regression equation?d. What is the value of R2 given in the output?e. Interpret R2 in the context of the problem.f. Using R2, the coefficient of determination, calculate the correlation, r. Round your answer to two decimal places.g. 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 the assumptions of linear regression, normality and constant variance. Let’s verify the assumptions. To test normality, we use a Probability Plot (or Q-Q plot). We can visually check for constant variance using a Residual vs Fits plot.i. First check normality. Copy and paste the probability plot. What conclusion can be made regarding this assumption?• 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.ii. Next, check constant variance. 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.) .Copy and paste your graph. What conclusion is made regarding the assumption of constant variance?h. 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.Copy and paste your output.i. Using the output, write the new regression equation (with outlier deleted).j. What is the new R2?k. Using the new R2, calculate the correlation between Math Score and IQ with the outlier removed. Round to two decimal places.l. 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. You should also repeat the residual plot and probability plot! Copy and paste your output.