AEMA 310_Final Fall 2022_FIN_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 <6> of 12 Statistical Methods 1 (AEMA 310) December 14, 2022 Part II (continued) S ECOND E PISODE (P ROBLEM 1): As a follow-up, Dr. Long wants to investigate whether the profitability of CSA farms (in $) is related to the amount of time (in hours) that participants spend working on the farm. He believes that when participants are more involved (i.e., spend more time working on the farm), it increases the profit. To verify this, he randomly samples 5 farms of similar size (i.e., land area, similar crops, number of participants). For each farm, he calculates the total profit for one year and the number of hours worked by participants that year. He obtains the following data: Total profit (,000 $) x i Amount of time worked (hours) y i (x i − x ̅) 2 (y i − y ̅) 2 (x i − x ̅)(y i − y ̅) 20 100 108.16 529 239.2 25 110 29.16 169 70.2 32 120 2.56 9 - 4.8 35 135 21.16 144 55.2 40 150 92.16 729 259.2 Ʃ = 152 Ʃ = 615 Ʃ = 253.2 Ʃ = 1580 Ʃ = 619 Ʃ/5 = 30.4 Ʃ/5 = 123 Ʃ/4 = 63.3 Ʃ/4 = 395 Ʃ/4 = 154.75 10) Is Dr. Long correct in his belief? Perform the appropriate test with  = 0.05. 9

Final Examination Page <9> of 12 Statistical Methods 1 (AEMA 310) December 14, 2022 Part II (continued) T HIRD E PISODE (P ROBLEM 1): Apple ice cider is a fermented beverage made from the frozen juice of apples. Cryo- concentration (i.e., the operation of freezing the juice of harvested apples, followed by a cold fermentation) and cryo-extraction (i.e., the extraction of juice from frozen apples left on the tree) are two approaches to producing ice cider. As the owner of an apple orchard, you want to produce ice cider with high sugar concentration at the time of fermentation, and you think that cryo-extraction would give better results. You therefore identify 5 trees of the same cultivar in your orchard. For each individual tree, you harvest half of the apples, extract the juice and freeze it. You later harvest the remaining apples once frozen on the trees and extract their juice. Before fermentation, you measure sugar content (in °Brix) of each juice. You obtain the following data and sample means: 1 2 3 4 5 Sample mean Initially harvested apples are used (cryo- concentration) 30 32 29 31 32 30.8 Apples left on the tree are used (cryo-extraction) 32 35 31 33 32 32.6 12) Based on the data above and the application of the appropriate test of significance, are you correct in thinking that cryo-extraction is a better way to produce ice cider (  = 0.05)? /9

Final Examination Page <12> of 12 Statistical Methods 1 (AEMA 310) December 14, 2022 Part II (last page) T HIRD E PISODE (P ROBLEM 4): To produce quality ice cider, it is recommended that the fermentation process be stopped when the concentration of residual sugar reaches 150 g/kg. If the concentration gets higher, the batch of fermented juice is thrown away because the taste of the ice cider would be affected otherwise … You want to know how many portions of fermented juice need to be analyzed in order to determine, with a probability of 0.95, that the mean concentration of residual sugar is above the recommended level while it is, in fact, 5 g/kg above it. Use  = 0.01, and you may assume that you somehow know the population variance is equal to 25 (g/kg) 2 . /5