Stat 501 Lab 031 Prediction and con…dence intervals for response variableThe exercise in this section is intended to review the methods we learned for predicting a new response Yand estimating the mean response E (Y ) for a given value of the predictor variable xh:1.1 Alcoholism and arm strength studyUrbano-Marquez et al. (1989) reported on strength tests for a group of 50 alcoholic men. Their daily intakeof alcohol ranged from 118 to 350 grams (with a mean of 243 grams) for an average of 16 years. The totallifetime consumption of alcohol (X = alcohol); in kg=kg of body weight, was determined for each personin the study. The response (Y = strength) was the strength of the deltoid muscle, in kg; in each person’snondominant arm. The response was determined by making …ve measurements over a 20-minute period,using an electric myometer, which measures muscular force against a …xed resistance. The resulting dataare stored in alcoholarm.txt.1. Using Minitab, determine a 95% con…dence interval for the mean arm strength of all alcoholic men whohave consumed a lifetime dose of 21 kg=kg of body weight of alcohol. Write a sentence that summarizesthe …ndings of the interval.2. Using Minitab, determine a 95% prediction interval for the arm strength of an alcoholic man who hasconsumed a lifetime dose of 21 kg=kg of body weight of alcohol. Write a sentence that summarizes the…ndings of the interval.3. Which interval – the prediction interval or con…dence interval – is wider? Will this always be the case?4. Would a prediction interval (or a con…dence interval) for a lifetime dose of 35 kg=kg of body weightbe wider or narrower than the above intervals? And, why? (Hint: To calculate the mean lifetimeconsumption of alcohol for the 50 men in the sample use Calc >> Column statistics.... SelectMean. Specify the input variable. Select OK.)5. Using Minitab, create a …tted line plot with 95% con…dence bands and 95% prediction bands. One ofthe data points (or observations) falls outside the prediction bands. Should this be surprising?2 Analysis of variance approach for testing H0: ¯1= 0The …rst exercise in this section is intended to review the analysis of variance (ANOVA) approach to testingfor a linear association between a predictor and a response. As you know, the ANOVA approach requires theuse of the F distribution. The second exercise is intended to help you understand what the F distributionis, and how it depends on the numerator and denominator degrees of freedom.2.1 Boiling point of water in the AlpsThe data set alpswater.txt contains the barometric pressure (in inches of mercury) and the boiling point(in degrees Fahrenheit) of water in the Alps. Treating the response Y = boiling and the predictor asX = pressure; and using Stat >> Regression >> Regression..., perform a basic regression analysis inMinitab.1. Using a signi…cance level ® = 0:05 and the P-value reported by Minitab that is associated with theF-test, what should we conclude?12. What relationship exists that guarantees that the P-value associated with the t-test will always be thesame as the P-value associated with the F-test? Con…rm that this relationship exists with this dataset.3. Use the F-test in conjunction with the critical value approach to test H0: ¯1= 0. Again, set ® = 0:05.Is your conclusion consistent with your conclusion made using the P-value approach?4. The regression sum of squares comprises what proportion of the total sum of squares? Does this …gureappear anywhere in the standard Minitab regression output?2.2 The F distributionIn this exercise, you are instructed to create one graph that contains two F distributions – the …rst onewith 10 numerator and 15 denominator degrees of freedom and the second one with 1 numerator and 24denominator degrees of freedom.1. For both of the F distributions, most of the F values fall between 0 and 3. So, let’s …rst createa column containing the F values. Label the …rst column (C1) ‘F’. Then, create the column of numbersbetween 0 and 3 by selecting Calc >> Make Patterned Data >> Simple set of numbers ... Now, …llin the boxes:² Store patterned data in: ‘F’² From …rst value: 0.1² To last value: 3² In steps of: 0.1² List each value: 1 time² List each sequence: 1 timeAfter selecting OK, ‘F’ should contain a set of numbers between 0.1 and 3, inclusive.2. Now, let’s create a second column containing the corresponding heights of the probability distributionof the …rst F distribution with 10 numerator and 15 denominator degrees of freedom. Label the secondcolumn (C2) ‘Ht1’. Then, select Calc >> Probability Distributions >> F... Select Probabilitydensity. In the box labled Numerator degrees of freedom, type 10. In the box labeled Denominatordegrees of freedom, type 15. Click on the button labeled Input column. In the box labeled Inputcolumn, select the variable ‘F,’ and in the box labeled Optional storage, select the variable ‘Ht1’. SelectOK. The variable ‘Ht1’ should now contain the height of the …rst probability distribution.3. Now, we just need to create a third column containing the corresponding heights of the probabilitydistribution of the second F distribution with 1 numerator and 24 denominator degrees of freedom. Label thethird column (C3) ’Ht2’. Then, select Calc >> Probability Distributions >> F... Select Probabilitydensity. In the box labled Numerator degrees of freedom, type 1. In the box labeled Denominatordegrees of freedom, type 24. Click on the button labeled Input column. In the box labeled Inputcolumn, select the variable ‘F,’ and in the box labeled Optional storage, select the variable ‘Ht2’. SelectOK. The variable ‘Ht2’ should now contain the height of the second probability distribution.4. Now, let’s plot the two F distributions on the same graph. Select Graph >> Plot... For Graph 1,select ‘Ht1’ as the y variable and ‘F’ as the x variable. For Graph 2, select ‘Ht2’ as the y variable and ‘F’as the x variable. In the area labeled Data Display, under Display, select Connect. Under Frame, selectMultiple Graphs... Under Generation of Multiple Graphs, select Overlay graphs on the samepage. Select OK. Select OK. The graph with the two F distributions should appear in a new window.5. Which of the two F distributions looks like the type of F distribution we will encounter when testingH0: ¯1= 0 in the simple linear regression setting?6. Use Minitab to …nd
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