AEMA 310_Final Fall 2022_FIN_2

Final Examination Page <1> of 12 Statistical Methods 1 (AEMA 310) December 14, 2022 Part I: Multiple-choice and true/false questions Multiple-choice questions ( 2 pts. each): Indicate clearly the most appropriate answer by filling in the box to its left. True/false questions ( 1 pt. each): Fill in one of the boxes to the left of each statement. If you believe the statement is T rue, then fill in the box below the T ; if you believe the statement is F alse, then fill in the box below the F . If you want to change a choice, erase your former choice and indicate the final one clearly. No mark will be given if two boxes or more are checked. 1) In the case of a simple linear regression, which of the following statements is correct ?  The expected value of Y for a given x can be calculated from the linear regression equation, as E(Y) = a + bx +  .  The ratio statistic ˆ b ˆ b/S has a “Student” t distribution with n – 2 degrees of freedom.  The estimate of the intercept must be equal to zero for the fitted linear regression model to be significant.  The value of the estimated slope is comprised between – 1 and 1 inclusively. 2) When testing H 0 :  1 2 =  2 2 against H 1 :  1 2 >  2 2 with n 1 = 6 and n 2 = 6, which of the following statements is correct ?  H 0 is rejected at level  when the observed test-statistic value is greater than χ 2 1-  (10).  H 0 is rejected at level  when the observed test-statistic value is greater than t 1-  (10).  H 0 is rejected at level  when the observed test-statistic value is greater than F 1-  (5, 5).  H 0 is rejected at level  when the observed test-statistic value is greater than z 1-  . 3) A researcher is aware that a concentration of 500 mg of heavy metals per kg of soil is the intervention threshold for decontamination. He wants to know how many samples would be required to detect with a 0.90 probability that the contamination level is above the threshold when it is in fact 50 above the threshold (  = 0.05 and  2 = 25). Which of the equations below can be correctly used for this purpose?  0.90 = (500 − 550)/√( 25 n ) + 1.282  1.282 = (500 − 550)/√( 25 n ) + 1.645 .  0.90 = (500 − 550)/ √( 25 n ) + 1.645  −1.282 = (500 − 550)/√( 25 n ) + 1.645 4) As a follow-up to Question 3), which of the following statements is not correct ?  Increasing  will increase the probability to detect a real difference between the acceptable concentration and the actual concentration of the heavy metal (Reject H 0 when it is false).  Lowering the power of the test will increase the possibility that decontamination will not be done while it should be.  Making a Type I error will cause unnecessary costs of decontamination.  Increasing  will lower the possibility that decontamination will not be done when necessary.

Final Examination Page <2> of 12 Statistical Methods 1 (AEMA 310) December 14, 2022 Part I (continued) 5) An experiment is conducted following an RCBD with one replicate per treatment per block, to test the effects of five treatments (for some disease) on five elderly men and five elderly women. In such an experimental situation, the Error df in the ANOVA decomposition is:  1  2  3  4 6) Consider an RCBD with p = 4, n = 4 and 1 replicate per treatment per block. Assume the conditions of application of a procedure of multiple pairwise comparisons of treatment means are satisfied. In this case, the LSD statistic (  = 0.05) is:  𝑡 0.95 (3)√𝐸𝑟𝑟𝑜𝑟 𝑀𝑆/2  𝑡 0.95 (3)√𝐸𝑟𝑟𝑜𝑟 𝑀𝑆/4  𝑡 0.975 (9)√𝐸𝑟𝑟𝑜𝑟 𝑀𝑆/4  𝑡 0.975 (9)√𝐸𝑟𝑟𝑜𝑟 𝑀𝑆/2

Final Examination Page <3> of 12 Statistical Methods 1 (AEMA 310) December 14, 2022 Part I (third and last page) 7) T F   The ANOVA model for an RCBD with more than 1 replicate per treatment per block is written as X ijk =  + a i + B j + (a i B j ) + ε ijk , where B j is the main effect of block j, a i B j is the effect of the treatment-by-block interaction, and both effects are fixed.   In the ANOVA model for an RCBD with 1 replicate per treatment per block, the main effect of treatment i (a i ) is fixed.   For a given confidence level, the length of the confidence interval for one population mean will decrease if the sample variance is increased while the sample size remains the same.   A Type II error is made when H 0 is rejected while, in fact, it is true .   When testing H 0 against H 1 in the probability approach, H 0 will be rejected if the probability of significance is smaller than  .   Increasing the sample size and the significance level lower the power of a statistical test.   When the population variance is known, the length of the confidence interval for the population mean is equal to 𝑧 1− 𝛼 2 √ 𝜎 2 𝑛 .   When testing H 0 :  a =  b against H 1 :  a >  b with known  a 2 and  b 2 , an effective number of degrees of freedom must always be calculated.   In a left-hand one-tailed test, the critical value of the test statistic decreases when  decreases.   If 1 bus accident happens every 3 weeks on average in a given part of the world, then the number of bus accidents in a 9-week period there follows a Poisson distribution X, with E(X) = Var(X) = 9.   An estimator of population parameter θ is said to be biased and imprecise if E( − θ) ≠ 0 and Var( ) is large.   If a 95% confidence interval for  is calculated to be [5, 6.5], with X  N(  ,  2 ), then the probability that  is less than 5 is 0.05.   If some H 0 is rejected against some H 1 at  = 0.01, then the same H 0 is automatically rejected against the same H 1 at  = 0.001.   When testing H 0 :  = 0 against H 1 :  ≠ 0, H 0 is rejected at the significance level  when the observed value of the test statistic, z obs , is smaller than – z 1-  /2 or greater than z 1-  /2 . ˆ  ˆ  ˆ 

Final Examination Page <4> of 12 Statistical Methods 1 (AEMA 310) December 14, 2022 Part II (in three “ episodes ”) F IRST E PISODE (P ROBLEM 1): Community-supported agriculture (CSA) farms are often synonymous with higher biodiversity because of the large number of crops grown in such systems. Dr. Small wants to investigate if this higher biodiversity helps with the amount of pollinator insects present in a production system. Note: A “healthy” production site (where pollination is adequate) is considered to have 6 pollinators per 9 m 2 on average during the day under some weather conditions (met here). 8-a) For a given production system, the producer tells Dr. Small that 80% of the flower clusters are pollinated adequately. In such a system, what is the probability that at least 2 flower clusters are pollinated adequately among 5 flower clusters randomly sampled. /5 8-b) At a “healthy” production site, Dr. Small evaluates an area of 6 m 2 , and count the number of pollinators. What is the probability that she counts at most 3 pollinators? /5