30) Building оn Stаnley Milgrаm’s study оn sоciаl networks, Duncan Watts and other researchers have found that
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Cоnsider dаtа оn X = Arm (upper аrm length in cm) and Y = Height (standing height in cm) fоr n = 75 individuals with height over 140 cm, randomly selected from the 2007-8 National Health and Nutrition Examination Survey. We would like to examine the relationship between these two variables. Minitab output for a simple linear regression model for this data is as follows: Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 53.84 9.70 5.55 0.000 Arm 3.043 0.257 11.85 0.000 1.00 Model Summary S R-sq R-sq(adj) 6.24897 65.79% 65.32% Analysis of Variance Source DF SS MS F P Regression 1 5482.85 5482.85 140.41 0.000 Error 73 2850.62 39.05 Total 74 8333.47 Graphical residual analysis resulted in the following: A t-distribution probability calculation resulted in the following: Student’s t distribution with 73 DF P( X ≤ x ) x 0.975 1.99300 Using the fitted model to make a prediction at X = 38 resulted in the following: Prediction Fit SE Fit 95% CI 95% PI 169.466 0.726508 (168.018, 170.914) (156.928, 182.004) Use the Minitab output to answer the following questions. Type your answers to the questions in the text box below, making sure to reference the relevant part of the output in each answer. (7 points) Report the fitted simple linear regression equation for these data and what does the scatterplot say about the relationship between two variables? (8 points) What are your impressions of the plot of the residuals versus the fitted values and the normal probability plot of the residuals. What do the plots tell us about our fitted model? (8 points) Test whether there is a statistically significant linear relationship at a 5% significance level. For full credit you must report the hypotheses, F-statistic, the degrees of freedom, the p-value and a conclusion in the context of the problem. (8 points) Calculate a 95% confidence interval for the slope parameter. Include details of the calculation and interpret the interval. (5 points) Report a 95% prediction interval for a new individual’s height for an upper arm length of 38 cm and interpret the interval. [There is no need to include details of the interval calculation.]