Statistics and Probability Answers

A sample of 81 observations has a mean 23 and standard deviation                                

 3.6, Test the Null Hypothesis H₀ : μ = 25 against the alternative

 H₁:  μ < 25, assuming α = .01

state the null

The lifetime T (years) of an electronic component is a continuous random variable with a

probability density function given by

( ) , 0 t

f t e t 

 

(a) Find the lifetime L which a typical component is 60% certain to exceed.

(b) If five components are sold to a manufacturer, find the probability that at least one of

them will have a lifetime less than L years

A study of four blocks containing 52 one-hour parking spaces was carried out and the results are given in the following table. Number of vacant one-hour parking spaces per observation period Observed frequency 2 3 26 31 45 20 15 7 3 Assuming that the data follow a Poisson distribution, determine: a) the mean number of vacant parking spaces, b) the standard deviation both ( i) from the given data and (ii) from the theoreti- cal distribution, and c) the probability of finding one or more vacant one-hour parking spaces, calculating from the theoretical distribution.

A sampling scheme for mechanical components from a production line calls for random samples, each consisting of eight components. Each component is classified as either good or defective. The results of 50 such samples are summa- rized in the table below. Number of Defectives Observed Frequency 30 17 From these data estimate the probability that a single component will be defective, Calculate the probabilities of various numbers of defectives in a sample of eight components, and prepare a table to compare predicted probabili- ties according to the binomial distribution with observed relative frequencies for various numbers of defectives in a sample.

An accounting company buys its computers from three different companies. Company X supplies 25% of the computers, company Y supplies 35% of the computers and company Z supplies the rest. From past experience you determine that 5% of company X’s computers produced are defective, 4% of company Y’s computers are defective and 3% of company Z’s computers are defective. One of the computers was reported as defective. By using Bayes' theorem or another method, what is the probability that the computer was supplied by Company X? 

Of the feed material for a manufacturing plant, 85% is satisfactory, and the rest not. If it is satisfactory, the probabifity it will pass Test A is 92%. If it is not satisfactory, the probability it will pass Test A is 9.5%. If it passes Test A it goes on to Test B; 99% will pass Test B if the material is satisfactory, and 16% will pass Test B if the material is not satisfactory. If it fails Test A it goes on to Test C; 82% will pass Test C if the material is satisfactory, but only 3% will pass Test Cif the material is not satisfactory. Material is accepted if it passes both Test A and Test B. Material is rejected if it fails both Test A and Test C. Material is reprocessed if it fails Test B or passes Test C. a) What percentage of the feed material is accepted? b) What percentage of the feed material is reprocessed? c) What percentage of the material which is reprocessed was satisfactory?

How many cups will likely overflow if 250 milliliters cups are used for the next 500 drinks?

Of the feed material for a manufacturing plant, 85% is satisfactory, and the rest not. If it is satisfactory, the probabifity it will pass Test A is 92%. If it is not satisfactory, the probability it will pass Test A is 9.5%. If it passes Test A it goes on to Test B; 99% will pass Test B if the material is satisfactory, and 16% will pass Test B if the material is not satisfactory. If it fails Test A it goes on to Test C; 82% will pass Test C if the material is satisfactory, but only 3% will pass Test Cif the material is not satisfactory. Material is accepted if it passes both Test A and Test B. Material is rejected if it fails both Test A and Test C. Material is reprocessed if it fails Test B or passes Test C. a) What percentage of the feed material is accepted? b) What percentage of the feed material is reprocessed? c) What percentage of the material which is reprocessed was satisfactory?

Define conditional probability. State and prove Baye’s theorem.