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Assignment 3 NAME 1. The fungus Neurospora crassa was grown at 30 degrees Celcius on an agar medium in tubes filled with an inert gas containing approximately 5%oxygen. The following growth rates in millimeters per hour were recorded for each inert gas used in the experiment. The molecular weight of each gas was also recorded as a predictor variable. The data are given in the following table. OBSERVATION 1 INERT GAS He MOLECULAR WEIGHT X 4.0 GROWTH RATE Y 3.51 2 3 4 Ne N2 N2 20.2 28.2 28.2 3.14 3.03 2.83 5 6 7 8 9 10 Ar Ar Kr Kr Xe Xe 39.9 39.9 83.8 83.8 131.3 131.3 2.71 2.76 2.27 2.17 1.88 1.85 In this experiment the exact value of the predictor variable, molecular weight, is essentially known. However, it is obvious that some experimental errors are associated with the measurement of the response variable, growth rate, since different values were observed for the growth rate when the same inert gas was used. If additional information could be obtained about the manner in which this experiment was performed, a list of the possible sources of error, or variation, in the observed responses could be constructed. It is a good idea to make such a list, but we will not do for this assignment. (a) Construct a graph of the data with the growth rates on the vertical axes. Does a straight line model appear to be appropriate for the relationship between the molecular weight of the inert gas and the growth rate of the fungus? . Before you proceed with this problem, use a straight edge to draw a line on the graph that appears to you to best fit the data. Use the line you have drawn on the graph to obtain “visual” estimates for the intercept and slope ntercept = slope = (b) Consider the simple linear regression model discussed in class in which the errors are assumed to be distributed as NID(0,2) random variables. Use this model in answering the rest of the questions. Find the least squares estimates for the parameters of the regression line. intercept = slope = What physical quantity does the slope represent in this case? (c) Test the null hypothesis that the slope is equal to zero against the alternative that it is not zero. Report the values for t-test = d.f.= p-value = State your conclusion. (d) Construct a 95% confidence interval for the intercept. lower limit= upper limit= Explain why this interval may be an unrealistic interval estimate for the growth rate when X is zero. (e) Estimate the average growth rate for a gas with a molecular weight of 50. Estimated average growth rate = Construct a 95%confidence interval for the average growth rate. lower limit= upper limit= (f) If the experiment was repeated using Helium (He) as the inert gas, estimate the growth rate that would be observed. Estimated growth rate = Construct a 95% interval for the potential value for the observed growth rate. lower limit = upper limit = (g) Complete the following AMOVA table : SOURCE OF VARIATION DF SUM OF SQUARES REGRESSION ON X EXPERIMENTAL ERROR CORRECTED TOTAL MEAN SQUARE F (h) Determine the proportion of variation in the observed growth rate that is explained by the simple linear regression model. R-squared = 2. Occasionally a linear regression model is considered in which the intercept is specified to be zero. This model is represented as Yi= Xi+ i , i = 1,2,…,n , where the i are NID(0, 2). Note that this regression line must pass through the origin, that is, the mean of the response is zero when X is zero. (a) Write down the quantity that must be minimized to find the least squares estimate of . (b) Write out the normal equation. (c) Find the solution to the normal equation. b= (d) Give a formula for the true variance of b. (e) How would you estimate 2=VAR( i) ? Give a formula. (h) Give a formula for a 95% confidence interval for . 3. Consider the simple linear regression model Yi= 0 +1Xi+ i , i = 1,2,…,n , for which the random errors are independent and identically distributed with a normal distribution with mean zero and variance 2. Show that the covariance _ between the sample mean of the responses Y and the least squares estimate for the slope of the regression line is zero. Which properties of the model are used in your proof?