- Please Answer Part A And B 1 (66.17 KiB) Viewed 21 times
PLEASE ANSWER PART (a) AND (b)
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PLEASE ANSWER PART (a) AND (b)
PLEASE ANSWER PART (a) AND (b)
Question 4 - Compulsory (33 marks) (a) Consider a sample of i.i.d. observations {X1,...,Xn} each with probability distribution N(4.1), where is unknown. (i) Provide an expression for the log-likelihood estimator of u. [5] (ii) From (a)(i), demonstrate that the log-likelihood estimator of p is the sample mean X [8] (iii) Under the assumptions stated above, justify why we may state that E[X] = #. [3] (b) Vitamin B6 is one of the vitamins in a multiple vitamin pill manufactured by a pharma- ceutical company. The pills are produced with a mean of 50 milligrams of vitamin B6 per pill. The company believes that there is a deterioration of 1 milligram per month, so that after 3 months it expects that y = 47. A consumer group suspects that the deterioration is greater than 1 milligram per month. A random sample of 20 pills were tested and yielded a mean of 46.94 milligrams of vitamin B6 per pill and a sample standard deviation of 0.15. (i) State with reason whether you would carry out a one-sided or a two-sided test. [3] (ii) State the null and alternative hypotheses. [3] (iii) Define a critical region to test the hypothese from (b)(ii) at the significance level a = 0.05. (iv) Compute the value of an appropriate test statistic. (v) Explain whether you would accept the null hypothesis and why. (3)
Question 4 - Compulsory (33 marks) (a) Consider a sample of i.i.d. observations {X1,...,Xn} each with probability distribution N(4.1), where is unknown. (i) Provide an expression for the log-likelihood estimator of u. [5] (ii) From (a)(i), demonstrate that the log-likelihood estimator of p is the sample mean X [8] (iii) Under the assumptions stated above, justify why we may state that E[X] = #. [3] (b) Vitamin B6 is one of the vitamins in a multiple vitamin pill manufactured by a pharma- ceutical company. The pills are produced with a mean of 50 milligrams of vitamin B6 per pill. The company believes that there is a deterioration of 1 milligram per month, so that after 3 months it expects that y = 47. A consumer group suspects that the deterioration is greater than 1 milligram per month. A random sample of 20 pills were tested and yielded a mean of 46.94 milligrams of vitamin B6 per pill and a sample standard deviation of 0.15. (i) State with reason whether you would carry out a one-sided or a two-sided test. [3] (ii) State the null and alternative hypotheses. [3] (iii) Define a critical region to test the hypothese from (b)(ii) at the significance level a = 0.05. (iv) Compute the value of an appropriate test statistic. (v) Explain whether you would accept the null hypothesis and why. (3)
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