Statistics and Probability Answers

It is the most common model for relative frequencies of a continuous random variable

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 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.

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.

Among 157 African-American men, the mean systolic blood pressure was 146 mm Hg with a standard deviation of 27. We wish to know if on the basis of these data, we may conclude that the mean systolic blood pressure for a population of African-American is 140.

For a sample of size 36, ∑x = 761.6, ∑x² = 16125.5. Is the population mean 21

From a box containing 4 black balls and 2 green balls, 3 balls are drawn in succession. Each ball is placed back in the box before the next draw is made. Let G be a random variable representing the number of green balls that occur. Find the values of the random variable G.

A word is to be formed using some or all of the 10 letters. M A N C H E S T E R Find the total number of ways of forming the word if a) The word is 10-letter long. The letters can be in any combinations. (2 marks) b) The word is 10-letter long. The word must begin with a consonant and end with a vowel. (5 marks) c) The word is 2-letter long, and must not include T or E. (2 marks) d) The word is 7-letter long. The word must contain at least one E. (6 marks) e) The word is 5-letter long, no other restrictions. (5 marks) 

The following data represent the number of hours that a rechargeable battery operates before a recharge is required: 1.5, 2.2, 0.9, 1.3, 2.0, 1.6, 1.8, 1.5, 2.0, 1.2, 1.7. Use the sign test to test the hypothesis at 0.05 level of significance that the battery operates with an average of 1.8 hours before requiring a recharge. State clearly your assumptions.

. To determine the effectiveness of a new safety control system in an industrial plant the number of accidents were measured for two weeks before and two weeks after its installation. The following data were obtained: 3 and 1 5 and 2 2 and 0 3 and 2 3 and 2 3 and 0 0 and 2 4 and 3 1 and 3 6 and 4 4 and 1 1 and 0 Use the paired sample sign test to test, at level 0.05, whether or not the new system is effective.

1. The number of accidents recorded on a freeway possesses a Poisson distribution with an average of 

three (3) accidents per week.

a. What is the probability that there will be no accident in a particular week?

b. What is the probability that there will be exactly five (5) accidents in a particular week?

c. Find the expected number of road accidents on the freeway per year if the weekly numbers of 

the recorded accident are independent.

2. A survey unofficially claimed that in every five (5) young executives, only one (1) practices good reading 

habits.

a. What is the probability that out of 10 young executives, two executives practice good reading 

habits?