Statistics Homework Questions and Answers

Question 1: Probability of Total Insurance Premium Alteration

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 mean 175 cm and standard deviation 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 predetermined mass of k g.

(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?