11 Problem 1 A random sample of 17 sophomores and a random sample of 13 juniors attending a large university yield the following descriptive results on grade point averages: 1 = 2.84, 1 = 0.52, and 1 = 2.98, 1 = 0.31. The normal probability plots suggest that the populations are normal. a) Are these samples independent or paired? b) Do the data provide sufficient evidence to conclude that the true mean GPAs of sophomores and juniors at the university differ? Test this at the 5% significance level. Include the hypotheses, case, test statistic, RR, p-value, degrees of freedom (if appropriate), decision, and conclusion in the context. c) Calculate and interpret an appropriate interval (or bound) and test the hypotheses in b). Explain how you test and write the decision. Problem 2 Repeat parts b) and c) of Problem 1 using the assumption of equal variances. Problem 3 Study the relationship between the height and weight of students at a large university. The scatterplot of weight against height is: The test of the correlation between weight and height: The qq-plot for residuals: The test about correlation between the fitted values and residuals: The scatterplot of residuals against fitted values: Below is the simple linear regression output from R: a) Are the assumptions for a simple linear regression met? Use the plots and correlation test output above to check the four assumptions. For each assumption indicate: which of the plots below (if any) you use and how, which of the correlation tests below (if any) and how, and whether the assumption is met? b) Based on the output, - report and interpret the estimated regression coefficients - write down the regression equation relating weight and height. - report the error, regression and total sums of squares and comment what the relationship between them is. - determine the residual standard deviation for this data set in three ways and comment what it estimates. c) Calculate and interpret a 95% confidence interval of the true slope and provide an interpretation of this interval. What is the sample size? d) Is height a significant linear predictor of weight? Use the output to test this at 0.05 level of significance. Explain how you test. Include your hypotheses, decision, and conclusion. e) Find the predicted weight for a person of height 70. f) Would it be appropriate to use the estimated regression model to predict the weight of a 5 year old? If yes, what is the predicted value? g) Report the coefficient of determination and give its interpretation as it relates to this study. Also find the value of the correlation coefficient between weight and height for this study. Explain how you found it. 3) i) We see that all the assumptions for simple linear regression are satisfied. From the first plot we see that the height and weight of bthe students are correlated and from the data below we see that the correlation coefficient is 0.7111, that indicates that the weight and height are strongly correlated. We can say that validity assumption. ii) In the scatter plot between height and weight all the points are around a straight line therefore we can say that we can say that the deterministic component (weight) is the linear function of the predictor(height). iii)In the scatter plot of residuals against fitted value we find the points are scattered everywhere which indicates that there is no correlation between the two. Therefore it indicates the independence of errors. iv) The Normal QQ plot indicates the equal variance of errors as well as normality of errors. b) coefficient of regression is 0.7111 The equation of the line of regression is - Y=5.488x 222.477 Error is 72.41 Sum of squares (height)=9715.1 Sum of squares (residuals)=9495.8 c) Confidence interval for the true slope is 0.3730769 to 0.3730769 Sample size =27 e) y=5.488*70 222.477= 161.68 hence the weight corresponding to height 70 is 161.68 f) It won't be appropriate as the lowest value of height is 62 . g) since it is a linear relation between two variables therefore coefficient of determination =(0.7111)^2= 0.506 This means that 50% of variation in y is predictable from x. Correlation coefficient between weight and height is 0.7111 3) i) We see that all the assumptions for simple linear regression are satisfied. From the first plot we see that the height and weight of bthe students are correlated and from the data below we see that the correlation coefficient is 0.7111, that indicates that the weight and height are strongly correlated. We can say that validity assumption. ii) In the scatter plot between height and weight all the points are around a straight line therefore we can say that we can say that the deterministic component (weight) is the linear function of the predictor(height). iii)In the scatter plot of residuals against fitted value we find the points are scattered everywhere which indicates that there is no correlation between the two. Therefore it indicates the independence of errors. iv) The Normal QQ plot indicates the equal variance of errors as well as normality of errors. b) coefficient of regression is 0.7111 The equation of the line of regression is - Y=5.488x 222.477 Error is 72.41 Sum of squares (height)=9715.1 Sum of squares (residuals)=9495.8 c) Confidence interval for the true slope is 0.3730769 to 0.3730769 Sample size =27 e) y=5.488*70 222.477= 161.68 hence the weight corresponding to height 70 is 161.68 f) It won't be appropriate as the lowest value of height is 62 . g) since it is a linear relation between two variables therefore coefficient of determination =(0.7111)^2= 0.506 This means that 50% of variation in y is predictable from x. Correlation coefficient between weight and height is 0.7111