Fall 2012 Points: 35 pts. Consider the following snip it from Section 3.4 of our textbook. Data Description

Transcription

1 STAT 360: HW #4 Fall 2012 Points: 35 pts Name: SOLUTION Consider the following snip it from Section 3.4 of our textbook. Data Description The data are haystack measurements taken in Nebraska in 1927 and 1928, when farmers sold hay unbaled and in the stack, requiring estimation of the volume of a stack. Two measurements that could be made easily with a rope were usually employed: the circumference around the base of the stack and the OVER, the distance from the ground on one side of a stack to the ground on the other side of the stack. Stacks vary in height and shape so using a simple computation like the volume of a hemisphere, while perhaps a useful first approximation, may or may not be sufficiently accurate. Source: Methods of Correlation Analysis, Ezekiel, M., (1941), 2nd ed, New York: Wiley, p Data Typical Haystack (I think ) 1

2 Answer the following using whatever software you d like. Marginal Distribution 1. Obtain the following quantities for Volume. (3 pts) Getting these quantities in JMP. Mean = 3,017.8 Variance = 836,672.5 Total Unexplained Variation = (given by Corrected SS value in JMP, can be computed as Variance * N as well) 2

3 Distribution of Volume Circumference 2. Create a plot of Volume vs. Circumference. Volume is the response variable for this investigation, so place this variable on the y axis in your plot. Give a brief statement (one or two sentences) about the general pattern you see in this plot. (3 points) Certainly, as Circumference increases so does Volume. However the trend/pattern does not appear to be linear. That is, the rate of change in Volume is greater for larger circumference values than for lower circumference values. 3. Show the math as to why it is reasonable to estimate the volume of a haystack using the quantity. (3 pts) Volume of sphere is given by and so the volume of a hemisphere is. Also, the circumference is given by 2. Solving 2 for yields and plugging this quantity into the volume of a hemi-sphere yields 3

4 Use the function above as an estimate of the mean function for Volume Circumference. Plot this estimated mean function on the graph created for Problem #2. Note: You may have to obtain an estimate of the mean function for each data point in the dataset in order to construct this plot. (3 pts) 4. Obtain Residual 2 value for each point in your dataset. Sum up these values to obtain the total unexplained variation for the mean function given above. (2 pts) Total Unexplained Variation =

5 5. What proportion of the total unexplained variation is being explained by using this mean function? Show the math here. (2 pts) Proportion = % Distribution of Volume Over 6. What would be a reasonable function for estimating volume if the Over measurement was used instead of the Circumference measurement? Explain how you obtained this value. (4 pts) If haystacks are a sphere, then the Over measurement would be equal to 2 as Over is the top half of the sphere. Thus, Use your function in Problem #7 to obtain the Residual 2 value for each point in your dataset. Sum up these values to obtain the total unexplained variation for the mean function using Over. (3 pts) Total Unexplained Variation = Once again, what proportion of the total unexplained variation is being explained by the using a mean function based on Over? (2 pts) Proportion = % 5

6 Comparing the Estimating Mean Functions 9. Which is a better estimate to use for estimating the average volume of a haystack the Circumference of the Over measurement? Explain. (4 pts) The Over measurement is better as the proportion of unexplained variation being taken away is somewhat higher for Over vs. Circumference (about 80% vs. 50%). It should be noted that using the Over measurement to predict Volume does have faults. Consider the fact that most residuals are negative using the Over measurement (see plots below). The practical ramifications of this would be that you believe you are buying more hay than you actually are; that is, the actual volume is less than is predicted by the Over measurement. Volume Circumference Volume Over 6

7 Investigation of the Variance Function 10. Pick one set of the residuals obtained above. Obtain the Residual value for each data point and plot it against Circumference or Over, depending on which mean function was used. (3 pts) Variance Function when using Circumference Variance Function when using Over 11. Discuss the general pattern seen in the above plot. Does the variance function appear to increase/decrease/not change as the haystacks get larger? Discuss the possible consequences of a changing variance function on someone who purchases hay in this manner. (3 pts) The variance function when using Circumference is more uniform and even. The amount that would be over-paid or under-paid is about equal when using Circumference. The Residual values appear to be somewhat smaller when using Over and increase slightly as a function of Over. That is, the amount of error in our prediction is larger for larger values of Over. One final comment when the Residual values are used we may mask certain patterns present in the residuals. This is the case in this example as most residuals were negative, but this information is lost when considering only the Residual values. A second plot of the actual residuals vs. Over would appropriate here. 7