HW#1

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Texas A&M University *

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Apr 3, 2024

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Farzaneh Hashemabadi HW#1 STAT 652-Spring 2024 UIN: 233005386 Email: Farzaneh.hashemabadi@tamu.edu Problem 1. Athletes are constantly seeking measures of the degree of their cardiovascular fitness prior to a major race. Athletes want to know when their training is at a level which will produce a peak performance. One such measure of fitness is the time to exhaustion from running on a treadmill at a specified angle and speed. The important question is then Does this measure of cardiovascular fitness translate into performance in a 19-km running race? 20 experienced distance runners who professed to be top condition were evaluated on the treadmill and then had their times recorded in a 10-km race. Run JMP to answer the following questions. You get the data from my website. (1) Look at the scatter plot of the data. Does a linear model seem appropriate? Re: Based on the scatter plot, the data points show significant deviations from the fitted line, indicating that a linear model might not accurately represent the dataset. Therefore, alternative modeling approaches should be considered to better capture the relationship between the variables. (2) Fit the linear regression model, 10 km time =B 0 + B 1 (Treadmill Time) + error. a) Report the estimated error variance. Is equal to MSE = 3.688 b) Report the 95% confidence interval for B 1 Re: CI B 1 = [-2.637, -1.308] c) Test H0: B 1 = 0. Did you reject H0?

Farzaneh Hashemabadi HW#1 STAT 652-Spring 2024 UIN: 233005386 Email: Farzaneh.hashemabadi@tamu.edu Re: Given the p-value, which is less than 0.0001, we can reject the H0, so B 1 ≠ 0 d) Test H0: B 1 = -2 (You are testing if the 10-km race time is reduced 2 minutes when you do 1 more minute in treadmill). Did you reject H0? Re: H0: B 1 = -2 H1: B 1 ≠ -2 The P-value is significantly lower than 0.05, so we can reject the null. B1 could not be -2. e) Is there a linear relationship between the amount of time needed to run a 10km race and the time to exhaustion on a treadmill? Be 95% confident. Re: Given the Correlation coefficient, and the CI correlation = [-0.92,-0.6]. It seems there is a linear relationship. f) Estimate the mean time to run 10km for athletes having a treadmill time of 11 minutes. Re: 10-km = 60.12501 - 1.9728355*Treadmill → if Treadmill = 11, 10-km = 38.45 g) Report a 95% confidence interval on the mean time to run 10km for athletes having a treadmill time of 11 minutes. Re: Mean Interval: [37.19956513, 39.648074593] h) Report a 95% prediction interval on the time to run 10km for an athlete having a treadmill time of 11 minutes. Re: Prediction Interval: [34.207324361, 42.640315363] i) Which of the following is a true statement? Be 95% confident. The confidence interval is wider than the prediction interval The prediction interval is wider than the confidence interval. (Correct) There are no differences between the prediction interval and the confidence interval.

Farzaneh Hashemabadi HW#1 STAT 652-Spring 2024 UIN: 233005386 Email: Farzaneh.hashemabadi@tamu.edu Problem 2 A manufacturer of cases for sound equipment requires drilling holes for metal screws. The drill bits wear out and must be replaced; there is expense not only in the cost of the bits but also for lost production. Engineers varied the rotation speed of the drill and measured the lifetime y (thousands of holes drilled) of four bits at each of _ve speeds x. Run JMP to answer the following questions. You get the data from my website. (1) Look at the scatter plot of this data (x versus y). a) Does there appear to be a relation? Yes, given the scatter plot there is a relation between X and Y. b) Does the relation appear to be linear? No, it does not look like to be fitted with a line, since only 39.1% variance of the data are captured by the linear model. (2) ii.Fit the linear regression model Lifetime = B0 + B1 _ (DrillSpeed) + error. a) What does the sign of the slope indicate about the relationship between the speed of the drill and lifetime? _ no relationship between lifetime and speed _ decreasing relationship between lifetime and speed (Correct) _ increasing relationship between lifetime and speed b) Would you report your results by only looking at the sign of the slope?

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Farzaneh Hashemabadi HW#1 STAT 652-Spring 2024 UIN: 233005386 Email: Farzaneh.hashemabadi@tamu.edu _ the slope is significant so YES (Correct) _ there does not seem to be a linear relationship so NO c) Predict values for x = 60; 80; 100; 120; 140. For which x values, most of the actual y values are larger/smaller than the predicted. (there is more than one answer so you need to check all possible answers to get the credit). Re: X= 60, Predicted y Value = 5.01 - For X= 60, all the actual y values are smaller than the predicted value. X= 80, Predicted y Value = 4.67 - For X= 80, all the actual y values are larger than the predicted value. X= 100, Predicted y Value = 4.33

Farzaneh Hashemabadi HW#1 STAT 652-Spring 2024 UIN: 233005386 Email: Farzaneh.hashemabadi@tamu.edu - For X= 100, most of the actual y values are larger than the predicted value, except one. X= 120, Predicted y Value = 3.99 - For X= 120, most of the actual y values are larger than the predicted value, except one. X= 140, Predicted y Value = 3.65 - For X= 140, most of the actual y values are smaller than the predicted value, except one. d) What does this pattern indicate about whether there is a linear relation between drill speed and the lifetime of the drill? Re: The observed pattern indicates that there may not be a linear relationship between drill speed and the lifetime of the drill. With an Rsquare value of 0.39, the linear model cannot

Farzaneh Hashemabadi HW#1 STAT 652-Spring 2024 UIN: 233005386 Email: Farzaneh.hashemabadi@tamu.edu adequately explain all variations in the data. This suggests that other factors besides drill speed likely influence the drill's lifetime, and a linear model may not be the most appropriate fit for the data. Further analysis or alternative modeling approaches may be needed to better understand the relationship between these variables.

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Farzaneh Hashemabadi HW#1 STAT 652-Spring 2024 UIN: 233005386 Email: Farzaneh.hashemabadi@tamu.edu Problem 3. A pharmaceutical _rm would like to obtain information on the relationship between the dose level and potency of a drug product. To do this, each of 15 test tubes is inoculated with a virus culture and incubated for 5 days at 30° C. Three test tubes are randomly assigned to each of the five deferent dose levels to be investigated (2, 4, 8, 16, and 32 mg). Each tube is injected with only one dose level and the response of interest (a measure of the protective strength of the product against the virus culture) is obtained. You get the data from my website. Run JMP to answer the following questions. (1) Plot the data, Response versus Dose level. Does linear regression appear to be appropriate? Since the Rsquare = 0.773, it means the linear regression model captured about 77% of variance of the data. It does not mean that the linear model is the most fitted model, a nonlinear one maybe map the data better. (2) Fit a linear regression model, Predicted response = B0 + B1 (dose level) + error. a) Report the estimated B0 and B1. B0= 8.67, B1= 0.57 b) Report the R square for this model. c) Use a residual plot to investigate a lack of fit. Which of the following is a correct statement according to a residual plot? _ The residual plot indicates that linear regression is a good fit. _ The residual plot indicates "lack of fit." (Correct) d) Conduct a test for lack of fit of the linear regression model. In this test, what is the value of the test statistics? What is the p value of this test? Re: H0 = There is no lack of fit H1 = There is a lack of fit F- statistic = 10.01 (from JMP) P-value = 0.0023 (from JMP) P-value < 0.05, so we can reject the null, it means that the linear regression model does not fit the data well, and there is a lack of fit. e) Is there sufficient evidence that the linear model is inadequate? Be 95% confident. Re: Yes, since the P-value < 0.05, we can reject the null, it means that the linear regression model does not fit the data well, and there is a lack of fit. (3) Fit a quadratic regression model Predicted response = B0 + B1 (dose level) + B2* (dose level) 2 . a) What is the estimated value for B0, B1 and B2? Re:

Farzaneh Hashemabadi HW#1 STAT 652-Spring 2024 UIN: 233005386 Email: Farzaneh.hashemabadi@tamu.edu B0 = 8.63 B1 = 0.83 B2 = -0.02 b) What is the Adjusted R Square value? Which model has a higher adjusted R square? The linear regression model or the quadratic regression model.

Farzaneh Hashemabadi HW#1 STAT 652-Spring 2024 UIN: 233005386 Email: Farzaneh.hashemabadi@tamu.edu Re: From JMP, Adjusted R Square = 0.86 for quadratic regression model. From JMP, Adjusted R Square = 0.75 for linear regression model. So the Adjusted R Square for quadratic regression model is higher than the one for linear regression model. c) Look at the residual plot. Which model appears to fit the data better? _ Quadratic regression model (Correct) _ Linear regression model _ Neither. The log of independent variable may help better than the quadratic model.

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