MTH220: An Insurance company Rounds 50000 insurance Premiums to the nearest Dollar: Statistical Methods and Inference Assignment, SUSS, Singapore

Question 1 An insurance company rounds 50000 insurance premiums to the nearest dollar. Assuming that the fractional parts of the premiums are continuous and uniformly distributed between 0 and 1, compute the probability that the total amount owing will be altered by more than $60. You may use R to assist you in answering this question, but you would need to list out all working details.

Question 2 A new drug cures 9 of 200 patients suffering from a type of cancer, for which the historical cure rate is 2%. Perform a test to check on the significance of this result, at both 5% and 1% levels of significance. Based on your conclusion, comments on the efficacy of the treatment using this new drug.

Question 3 Men’s heights are normally distributed with mean 180 cm and standard deviation 4 cm, while women’s heights are normally distributed with a mean of 175 cm and standard deviation of 3 cm. In a sample of 100 married couples, the average difference between husband’s height and wife’s height is 3.4 cm. By performing an appropriate test, comment on if the choice of partner in marriage is influenced by considerations of height, at 1% level of significance. Show full working details for your test.

Question 4 Cans of black olives are filled by two machines (A and B) in a food processing factory. The distribution of the gross mass is known to be normal with mean 500 g and standard deviation 5.3 g for machine A, and normal with mean 504 g and standard deviation 4.8 g for machine B. A batch of canned black olives have been filled by the same machine, but it is not known which machine was used. As some substandard olives have accidently been used by machine A, it is decided to test H0: batch is from machine A, against H1: batch is from machine B By weighing a random sample of 6 cans and rejecting H0 if the mean mass exceeds a pre-determined mass of kg.

(a) Determine constant k such that the risk of type I error is 5%. What is the corresponding risk of type II error?

(b) Apply and carry out the test for a sample of 6 cans with masses 511, 499, 500, 498, 507, 495 g.

(c) It is decided to increase the sample size before reaching a final conclusion. Determine how many more cans are needed to be weighed if the risk of type I error is to remain at 5%, but the risk of type II error is to be reduced to just under 1%? What is the decision rule in this case?

Question 5 Professor Albert Francis had students in his Psychology 201 class rate his performance. The ratings are kept from the professor till the release of the course marks. The course marks are shown in the table below, under the respective rating given by the students. Performance Rating Excellent Good Fair Poor 94 75 70 68 90 68 73 70 85 77 76 72 80 83 78 65 88 80 74 68 65 65 Table Q5 Apply and perform an appropriate test to check if there is a difference in the mean marks received by students in each of the rating categories. Show all relevant workings and steps in the test, and test at 1% level of significance.

Question 6To explore the relationship between the weekly production of an employee and his/her years on the job, the following set of sampled data is obtained (Table Q6).