Midterm Review Q2-3

University of Calgary Department of Mathematics and Statistics Statistics 429 Midterm Examination Instructor: Dr. B. Cindy Sun Nov. 1, 2022 Time: 2:00-3:15pm Total Marks: 37 I.D. NUMBER GIVEN NAME(S) LAST NAME EXAMINATION RULES 1. This is a closed book exam. 2. Students arriving late will not normally be admitted after one-half hour of the examination time has passed. 3. No candidate will be permitted to leave the examination room until one-half hour has elapsed after the opening of the examination, nor during the last 15 minutes of the examination. All candidates remaining during the last 15 minutes of the examination period must remain at their desks until their papers have been collected by an invigilator. 4. All enquirers and requests must be addressed to supervisors only. 5. Candidates are strictly cautioned against: (a) speaking to other candidates or communicating with them under any circumstances whatsoever; (b) bringing into the examination room any textbook, notebook or memorandum not authorized by the examiner; (c) making use of calculators and/or portable computing machines not authorized by the instructor; (d) leaving answer papers exposed to view; (e) attempting to read other students’ examination papers. The penalty for violation of these rules is suspension or expulsion or such other penalty as may be determined. 6. Candidates are requested to write on both sides of the page, unless the examiner has asked that the left half page be reserved for rough drafts or calculations. 7. Discarded matter is to be struck out and not removed by mutilation of the examination answer book. 8. Candidates are cautioned against writing in their answer books any matter extraneous to the actual answering of the question set. 9. A candidate must report to a supervisor before leaving the examination room. 10. Answer books must be handed to the supervisor-in-charge promptly when the signal is given. Failure to comply with this regulation will be cause for rejection of an answer paper. 11. If a student becomes ill or receives word of domestic a ✏ iction during the course of an examination, he/she should report at once to the Supervisor, hand in the unfinished paper and request that it be canceled. Thereafter, if illness is the cause, the student must go directly to University Health Services so that any subsequent application for a deferred examination may be supported by a medical certificate. An application for Deferred Final Examinations must be submitted to the Registrar by the date specified in the University Calendar. Should a student write an examination, hand in the paper for marking, and later report extenuating circumstances to support a request for cancellation of the paper and for another examination, such request will be denied. 12. Attempt all questions.

2 . Many di ↵ erent interest groups — such as the lumber industry, ecologists, and foresters — benefit from being able to predict the volume of a tree just by knowing its diameter. One classic data set (shortleaf.txt) — reported by C. Bruce and F. X. Schumacher in 1935 — concerned the diameter (x, in inches) and volume (y, in cubic feet) of n = 70 shortleaf pines. https://online.stat.psu.edu/stat462/node/154/ ( Total 7 marks ) leaf <- read.table ( "shortleaf.txt" , header =T) attach (leaf) range (Diam) ## [1] 4.4 23.4 range (Vol) ## [1] 2.0 163.5 m1 <- lm (Vol ~ Diam) summary (m1) ## ## Call: ## lm(formula = Vol ~ Diam) ## ## Residuals: ## Min 1Q Median 3Q Max ## -18.899 -4.768 -1.438 6.740 45.089 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -41.5681 3.4269 -12.13 <2e-16 *** ## Diam 6.8367 0.2877 23.77 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 9.875 on 68 degrees of freedom ## Multiple R-squared: 0.8926,Adjusted R-squared: 0.891 ## F-statistic: 564.9 on 1 and 68 DF, p-value: < 2.2e-16 nDiam <- log (Diam) nVol <- log (Vol) m2 <- lm (nVol ~ nDiam) summary (m2) ## ## Call: ## lm(formula = nVol ~ nDiam) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.3323 -0.1131 0.0267 0.1177 0.4280 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -2.8718 0.1216 -23.63 <2e-16 *** ## nDiam 2.5644 0.0512 50.09 <2e-16 *** ## --- 5

(b) Decide which model is better for inference, explain how you decide which model is better. ( 2 mark ) (c) Interpret the slope estimate in the better model. ( 2 marks ) (d) Use the better model to predict volume of trees with diameter of 10 inches. ( 1 marks ) 7

3 . The data used below records height of a shrub (height) based on the amount of bacteria in the soil (bacteria) and whether the shrub is located in partial or full sun (sun). Height is measured in cm. Bacteria=1 if there are 100 thousand bacteria per ml of soil, and bacteria=0 if there are no bacteria in the soil. Sun=0 if the plant is in partial sun, and sun=1 if the plant is in full sun. ( Total 11 marks ) shrub <- read.csv ( "shrub.csv" , header =T) attach (shrub) range (height[bacteria == 1 ]) ## [1] 2.4 6.1 range (height[bacteria == 0 ]) ## [1] 1.9 5.3 range (height[sun == 1 ]) ## [1] 4.0 6.1 range (height[sun == 0 ]) ## [1] 1.9 3.8 bacteria = as.factor (bacteria) sun = as.factor (sun) m1 <- lm (height ~ bacteria * sun) summary (m1) ## ## Call: ## lm(formula = height ~ bacteria * sun) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.75000 -0.38333 0.04167 0.43750 0.76667 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 2.4333 0.1957 12.435 7.23e-11 *** ## bacteria1 0.6000 0.2767 2.168 0.0424 * ## sun1 2.3167 0.2767 8.371 5.74e-08 *** ## bacteria1:sun1 0.1167 0.3914 0.298 0.7687 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.4793 on 20 degrees of freedom ## Multiple R-squared: 0.8881,Adjusted R-squared: 0.8713 ## F-statistic: 52.9 on 3 and 20 DF, p-value: 1.08e-09 m2 <- lm (height ~ bacteria + sun) summary (m2) ## ## Call: ## lm(formula = height ~ bacteria + sun) ## 8

2.5 3.0 3.5 4.0 4.5 5.0 5.5 − 1.5 − 0.5 0.5 1.5 m1 fitted values Standardized residuals 2.5 3.0 3.5 4.0 4.5 5.0 5.5 − 1.5 − 0.5 0.5 1.5 m2 fitted values Standardized residuals (a) Write down models ‘m1’ and ‘m2’. ( 2 marks ) (b) Write down the null and alternative hypotheses involved in the ‘anova(m1,m2)’ output, and make a conclusion for this test. ( 3 marks ) 10

(c) Decide which model is better for inference, explain how you decide which model is better. ( 2 mark ) (d) Based on the better model, explain the influence of the predictors on the response. ( 2 marks ) (e) Use the better model to predict height of shrub with full sun and 100 thousand bacteria per ml of soil. Use the better model to predict height of shrub with partial sun and 0 bacteria per ml of soil. ( 2 marks ) 11