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5. Refer to the effect of packaging on sales example in Lecture note 3 and 4. Suppose you are interested in testing whether there is any difference in sales for package designs D3 and D4. Suppose also that σ-10 (a) Write down the null and alternative hypotheses (b) Write down the test statistic. Compute the P- value of the test (c) What is your conclusion (assume α 0.01 level of significance) ? 6. Suppose that, X1,… , xn are i.id. radon variables and iet, x = n Σ_i Xi (a) Show that the surn of residuals always equal to zero, i.e., show that Σ(X, X)-0 i-1 (b) Show that に! (c) Please answer the following questions and provide a brief explanation to your answers. What is the degree of freedom of (X1, . .. , xn) ? -What is the degree of freedorn of (x, , . . . , Xn, X) ? What is the degree of freedom of the residuals: (Xi - X,-.-. , Xn - X)? -What is the degree of freedom of (Xi X, . . . , X,-X, X) ?

Answer 1

From the given information, Suppose X, ,X, X. are iid. random variables and X ΣΧ, i-l cx i I

に! i=1 ア2 Degrees of freedom, Degrees of freedom is defined as the number of independent values assigned to a distribution The degrees of freedom of (X,X,X),n. Here, the number of independent values assigned to a distribution is n-1 .