MathScore IQ 33 31 35 38 41 37 37 39 43 40 41 44 40 45 48 45 31 47 43 48 Abstract_Reas 95 100 100 102 103 105 106 106 106 109 110 110 111 112 112 114 114 115 117 118 28 24 29 30 33 32 34 36 38 39 40 43 41 42 46 44 41 47 42 49 Lab Activity 11 - Simple Linear Regression 1. 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. Relationship between Height and Weight. Data has been collected on 225 STAT 200 students. Weight is measured in pounds and Height in inches. 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. Relationship between eighth grade IQ and ninth grade math score. For a statistics class project, students examined the relationship between x = 8th grade IQ and y = 9th grade math scores for 20 students. The data are displayed below. Student Math Score IQ Abstract Reas 1 33 95 28 2 31 100 24 3 35 100 29 4 38 102 30 5 41 103 33 6 37 105 32 7 37 106 34 8 39 106 36 9 43 106 38 10 40 109 39 11 41 110 40 12 44 110 43 13 40 111 41 14 45 112 42 15 48 112 46 16 45 114 44 17 31 114 41 18 47 115 47 19 43 117 42 20 48 118 49 Open the dataset IQ found in the Datasets folder in ANGEL. 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 Express: Graphs > Scatterplot > Simple. b. Perform a linear regression with the Response (dependent variable) math score and the variable IQ as the Predictor (independent variable). Click on the Graphs tab, and check the box for Residual plots. You'll use these plots in the next part. 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 Versus Fits Plot and test for normality using a Normal Probability Plot (or Q-Q plot). These plots were created during your analysis if you checked the box for residual plots. 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? 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. You may want to repeat the residual plot and probability plot