Stat 501 L ab 06The exercises for this lab is intended to review many of the simple linear regression concepts that we’velearned.1 Boyle ’s law a nd the l a d d e r of power sRobert Boyle examined the relationship between the volume in which a gas is contained and the pressurein that container. He used a cylindrical container with a moveable top that could be raised or lowered tochange the volume. He measured the height by counting equally spaced marks on the cylinder, and measuredthepressureininchesofmercury.Someofhisdataarerecordedinboyleslaw.txt .We’ve learned that the most common transformation of the response is the power transformation:y0= yλwhere λ (“lam bda”) is generally some meaningful number between −1 and 2. The resulting transformationsof the response, which are called the ladder of powers,are:λ −1 −12“0”121 2yλ1y1√ylogey√y y y2This exercise is intended to illustrate how as you move up or down the ladder of powers, you accomplishmore or less straightening of a curved relationship. The general idea is that if we are moving in the rightdirection but have not had sufficient effect, we should go farther along the ladder. On the other hand, ifwe’ve made the relationship more curv ed we should turn around and go in the other direction.1. Create a fitted line plot of y = pressure and x = height to see that indeed the relationship is curved.2. Try going up the ladder by creating a fitted line plot of y0= pressure2and x = height. Did thetransformation make the relationship more or less curved? That is, should we continue up the ladder, orshould we turn around and go in the other direction?3. Now, try going down the ladder by creating a fitted line plot of y0=√pressure and x = height.Did the transformation make the relationship more or less curv ed? That is, are we now going in the rightdirection?4. Continue down the ladder by creating a fitted line plot of y0=loge(pressure) and x = height. Is therelationship continuing to get less curved?5. Continue down the ladder by creating a fitted line plot of y0=1√pressureand x = height. Is therelationship continuing to get less curved?6. Continue down the ladder by creating a fitted line plot of y0=1pressureand x = height. Can it getan y better than this?Incidentally, recall that Boyle’s Law states that for a constant temperature, volume is inversely related to thepressure by the relationship V =αP, where α depends on the gas. T hat’s what we’ve shown in our analysisof his data where P denotes pressure and V denotes the volume (or height in this case) of the container.2 Length and weight of alligatorsWildlife biologists can fairly accurately determine the length of an alligator from aerial photographs or froma boat. Determining the weight of an alligator from a distance is much more difficult. Wildlife biologistsin Florida captured 25 alligators in order to collect data and to develop a model from which weight can bepredicted from length. The data set alligator.txt contains the resulting 25 measurements, x = length andy = weight.1. Fit a simple linear regression model with x = length and y = weight. In doing so, request theappropriate residual plots to iden tify everything that is wrong with the model. Based on the residual plots,what kind of transformation — on x? on y? or on both x and y? — would be most appropriate?12. Use the Minitab calculator to create a transformed predictor — the logeof the predictor x = length.Fit a simple linear regression model with x0=loge(length) and y = weight, and using residual plots, checkto see if the model assumptions are now met. Also, request a lack of fit test — does the conclusion suggestnonlinearity? If so, does it tell you what kind of function might be appropriate? Should you be surprised atthe results of your residual analysis?3. Now, use the Minitab calculator to create a transformed response — the logeof the response y = weight.Fit a simple linear regression model with x = length and y0=loge(weight) , and using residual plots, checkto see if the model assumptions are now met. Also, request a lack of fit test — does the conclusion suggestnonlinearity?4. For the simple linear regression model with x = length and y0=loge(weight), notice that Minitabhas flagged a data point as potentially having “large influence” on the estimated regression line:Unusual ObservationsObs length loge_wt Fit SE Fit Residual St Resid3 147 6.4615 6.5415 0.0981 -0.0800 -0.71 X18 61 3.7842 3.4957 0.0468 0.2885 2.04RR denotes an observation with a large standardized residualX denotes an observation whose X value gives it large influence.If y ou haven’t already done so, create a fitted line plot for the model with x = length and y0=loge(weight). Identify the point that Minitab has flagged. Is it believable that this data point can have alarger impact on the estimated line than some other data point say where length is 75? (We will discussinfluential data points later in the course; for now , beware that they exist and can heavily influence theestimated line.)5. Also, try fitting a simple linear regression model with x0=loge(length) and y0=loge(weight) , andusing residual plots, check to see if the model assumptions are met. Also, request a lack of fittest—doesthe conclusion suggest nonlinearity?6. Which of the three models do you think is best in terms of meeting the model assumptions? In ligh tof what was wrong with the original model with x = length and y = weight, does it make sense that thismodel should work? Use this model to predict the weight, with 95% confidence, of an alligator which is 100inch es long.3BodyfatdataThe bodyfat.txt data set contains data on the body fat (in percent), waist (in inches), and weight (inpounds) of 20 different individuals. The investigators are interested in answering two questions:• Do these data indicate an association between waist size and body fat?• Do these data indicate an association bet ween weight and body fat?Analyze the data to help answer the investigators’ questions.1. Fit a simple linear regression model with x1= waist and y = bodyfat. Check the model assumptions.Test an appropriate hypothesis about the association and state your conclusion. Quantify the strength ofthe association. Estimate, with 95% confidence, the mean percent body fat found in people with 40" waists.2. After having “removed” the variability in y = bodyfat “due to” x1= waist, does it appear as ifadding x2= weight to the
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