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

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