Question

A farmer is investigating the effect of a pesticide treatment on her barley crops. She observes the yields of barley on 6 equally sized fields, three of which are treated using a fertiliser. 3. Fields Untreated Treated Yields 35, Y12-37, Yİ3 31, Y22-39, Y 3 Y11 42 Y2 32 , 23 You have been asked to assess whether the treatment is effective (that is, whether yields have significantly increased as a result of the treatment). You may assume that the yields on different fields are independent normal random variables with common standard deviation σ and means μι (for untreated fields), μ2 (for treated fields). She wishes to test the null hypothesis Ho : U2- /1 against the alternative hypoth- esis that the pesticide treatment increases yields HA : μ2 11. (a) Estimate the mean yield, separately for untreated and treated fields.[2] (b) Calculate the residual sum of squares 2 (c) State (without proof) the theoretical distribution of the residual sum of squares, and explain any degrees of freedom calculation (d) Calculate the theoretical distributions of the following quantities: 8 (iii) Z3 -Yı2 - Y (e) Demonstrate that the following six quantities (the first four are the same as in question 3d) are independent. (f) Derive the distribution, under the null hypothesis, of the following quantities: [2 [21 [2 (g) Calculate the quantities in 3f using the data at the start of this question. [4] (h) Hence perform a one-sided 95%-test of whether the two means are significantly (iii) Z2 24 24 different. 2

Answer 1

(a) Estimate of the mean yield for untreated field Estimate of the mean yield for untreated fielod Y2- (31 +39 32)/3 34 (b) Residual surn of square-SSR = (3-1) (s-s2/ = 2 * (13+19) = 64 si - sample variance for untreated 13 s2sample variance for treated 19 (c) SSR ~ σΎί i.e. σ2 × X2 distribution with 4 df (d) (i) Z1 ~ Ņ (3411+p12), 6σ2): since Z1 is a linear combination Y s where Yi s are independently Ņ(μί.02) random variables (ii) Z2 ~ Ņ (3412-11), 6ơ2) since Z2 is a linear combination Yii s Where Yii s are independently Ņ( .02) random. variables (iii) Z3 ~ N(0.2ơ2) 2, 2 ㄅ 2Y13-Y11-Y12 4

(f) (i) Z2N6 ~ Ņ(0.gt) under Ho : μ,-μ2 6 t4 under Ho