Math 11 Lab #3

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Dec 6, 2023

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Lab #3 “Mammals” Regression of basal metabolic rate against body mass 1. What is the equation of the regression line? Provide a scatterplot with the line included. Line of Regression: y= 75.97+ 0.2822(x) , where x is the mass 2. What does the regression line predict for the basal metabolic rate of the Cape Porcupine? What about the San Diego Pocket Mouse? How do your predictions compare with the observed values? - The regression line predicts that the basal metabolic rate for the Cape porcupine is: y=75.97+0.2822(11300) -> 3264.83 - The regression line predicts that the basal metabolic rate for the Cape porcupine is: y=75.97+0.2822(19.6) -> 81.50112 - My predictions look like they do not match the actual BMR that was observed, because with the Cape Porcupine, we predicted 3264.83, but the observed was 2361.7. This isn't correct at all. We overshot. With the San Diego pocket Mouse, we predicted 81.50112, but the actual was 26.9. Also very wrong, and very high. It appears that our predictions tend to overshoot significantly.

3. Judging from the scatterplot and the residual plot, do you think that your linear regression model is appropriate for predicting basal metabolic rate from body size? Explain your answer, and show the necessary plots. - The residual plot shows no curvature, but rather shows heteroskedasticity. There are outliers in the scatter plot as well as the residual plot which mess with the slope and overall regression line equation. But, I would say that regression does seem appropriate, as the regression line shows us an overall, but still wrong when looked at closely, linear trend in the information provided. As for the outliers, we are able to see that they show up in both the scatter plot and the residual plot, which is good because they're consistent. Transforming the variables When working with highly skewed data, or scatterplots that show curvature or an uneven spread around the regression line, it is often useful to transform the variables. For this data set, try taking logarithms of both variables. This will require defining two new variables, one for the log of body mass and one for the log of basal metabolic rate. To define a variable for the log of body mass, first go to Calc --> Calculator . Under "Functions", scroll down to "Natural log" and double click on it. Then double click on "Mass" in the variables box. You should now see LN('Mass') in the box for Expression. Type in a name for your new variable in the box labeled "Store result in variable", and then click "OK". Then do the same for the basal metabolic rate. Now you can work with the two new variables. 4. Compare the histograms of the two original variables with the histograms of the two new transformed variables. Does taking logarithms of the variables reduce or eliminate the skewness? - Yes, taking the logarithms of the variables do reduce/eliminate the skewness. It simply removes it though, we are still able to see that it's skewed to the left. 5. Find the equation of the regression line using the transformed variables. - The equation of the regression line with the transformed variables is : Y = 0.1944+3.049x, where x is the mass 6. Do your results lend support to either of these two values? - I believe that my results lend more support to the value that b=¾. When inputting a random value from the chart and plugging in the mass and bmr where indicated, the numbers were closer to ¾.

7. Based on the scatter plot and residual plot, do you think that this regression model is appropriate? Which of the two models (this one, or the one that you considered in question 1) do you think is better? Explain your answers. - I think it is appropriate and this model is the better of the two. Visually, it looks significantly better because the data isn't all clumped together in one corner, which is insanely skewed, and rather it is evenly spread out throughout the regression line. 8. What does this model predict for the basal metabolic rates of the Cape Porcupine and the San Diego Pocket Mouse? How do these predictions compare with the observed values? Are these predictions more accurate than those obtained from the first model? - These predictions are now too low. They are about half of what the observed values are. Using the formula,BMR = c *Mass b , I was able to get the expected bmr value for the porcupine at 7.767, when the predicted was 8.27. Observed for the mouse was 3.29 and predicted was 4.308. These predictions seem to be more accurate than the ones obtained from the first model, because these are way closer than the first. These predictions seem to be off by 1 value, rather than ⅓. 9. What does your model predict for the basal metabolic rate of a Blue Whale? How much confidence do you have in this prediction? - I got the BMR of 3703.26 when basing it on the equation of BMR = c *Mass b . I am fairly confident in this prediction because all the others have proven to be close enough to the observed values. Comparing different types of mammals 10. Find an equation for predicting basal metabolic rate from body mass for rodents. Examine a scatter plot and residual plot to make sure linear regression is appropriate. Include in your report these plots and a brief discussion of whether or not linear regression is appropriate. - It is not appropriate based on the residual graph. Residual graphs are supposed to be uneven, random, scattered. This one shows order and a pattern. It is not appropriate. The equation for these rodents is -659.9+187.2(mass).

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11. How does the allometric exponent that you estimate for rodents compare to the exponent that you estimated for all mammals combined? - The allometric exponent that I estimated for the rodents in comparison to all mammals combined is interesting, because I see that the slope for this graph is 187.2, when with all mammals it was 263.3. We also see a difference in the y intercept, where the rodents have a smaller y intercept. Where with rodents it's -659.9 and with mammals it is -930.8. 12. Next, consider the primates (Order Primates ). Answer the same questions that you answered for the rodents. The residual graph shows more promise than the rodent, but it still is not appropriate because there isn't enough randomness and unevenness on the residual graph. We are able to see that the slope is 669.7, which is great for all mammals because with all mammals it was 263.3. We also see a difference in the y intercept, where the primates have a significantly larger y intercept. Where with primates its -3553 and the mammals is -930.8. 13. Finally, consider the bats (Order Chiroptera ). Answer the same questions that you answered for the rodents and the primates. - With bats, we are able to see that this residual graph isn't promising and cannot be used because again, not random enough. Very linear, although curved. We see that the slope is 69.42, which is smaller than the all mammals slope, which shows smaller increasing points, and the starting y value/intercept is smaller than the all mammals, where -163.8<930.8.

14. Write a paragraph consisting of several sentences summarizing your conclusions about the relationship between body mass and basal metabolic rate for mammals in general and for the three groups of mammals that you studied in more detail. The relationship between the body mass and the BMR for mammals seems to be fairly linear, with there being obvious outliers in the graph, but they're all following the same pattern, this includes the ln graphs. When we look at the individual groups, including bats, primates, and rodents, we see that these ln graphs tend to be more curved and they dont look as linear as they possibly could be. We see that there is a pattern but just not linear. None of the residual graphs proved to be fit to use as they were not random enough, the mammals one was cone shaped, and the closest to being fit to use was for the primates, which still didn't have enough randomness to be deemed fit to use for data analysis usage.

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