Part I. In a certain assembly plant, three machines, AA, BB, and C, make 15%, 19%, and 66%, respectively, of the products. It is known from past experience that 3%, 5%, and 1% of the products made by each machine, respectively are defective. Please show work. No work no credit.
a) Draw a tree diagram of the scenario.
b) If a product is chosen at random, what is the probability that it is defective?
c) Given a randomly selected product was found to be defective, what is the probability it was made by machine A?
Part II. Match each distribution, and explain why, to each of random variable scenarios provided below.
A. Binomial Distribution B. Poisson Distribution C. Normal Distribution
D. Exponential Distribution E. Uniform Distribution
a) The chance a sample of a particular solution is contaminated is 0.09. Suppose 100 samples are tested for contamination. The probability one sample is contaminated is independent of the other samples. What type of distribution can be used to model the probability of contaminated samples?
b) The weight in ounces of certain bag of potato chips has a distribution that is symmetric, with an average of 12 ounces and a standard deviation of 0.5 ounces.
c) The distribution of random numbers produced from a computer is symmetric from 1 to 100 with every number in between have an equal likelihood of occurring. The outcome of the random number generator is a continuous.
d) The number of cars per hour at an automatic carwash has a discrete probability distribution with an average of 9 cars.
e) The time until replacement of a specific refrigerator under normal conditions is a continuous random variable with and expected lifetime of 11.1 years. The distribution is positively skewed.
Part III. Show all work! No work no credit.
Let X denote the length of time, in hours, that a statistics reference book on a three hour reserve at the engineering library is checked out by a randomly selected student. X is approximated by a continuous distribution with a pdf of:
f(x) = {c(1+x)-1 , 0≤x≤4
{0 , otherwise
a) Find c such that f(x) is a legitimate probability density function. Must show all work!
b) What is the expected time in hours that the book will be checked out? Find E(X). Must show all work.
c) What is the standard deviation of time in which the book will be checked out? Show work.
d) For books returned after 3 hours there is a $4 fine. What is the probability a randomly chose student will be charged a fine?
Part IV. Show all work! No work no credit.
The safety manager at a chemical facility knows that the time between accidents (in days) can be modeled by an exponential distribution. From historical data, he calculated that the mean time between accidents is 18 days.
a) What is the cumulative density function, F(X) = P(X
b) Calculate the median time between accidents in days? Why does it makes sense that the mean is greater than the median for this type of distribution?
c) What is the probability that the time between accidents will be greater than 30 days?
Part V. Show all work! No work no credit.
Historically, yield strength (in ksi) for specimens of A36 grade steel have been normally distributed with mean 45 ksi and standard deviation 3 ksi.
a) By hand or using R. See code under Data Analysis #2 Instructions. Draw the normal distribution, label the mean, ±1 σ, ±2 σ, and ±3 σ.
b) Using the 68-95-99.7 rule. What is the approximate proportion of specimens between 39 ksi and 51 ksi?
c) What is the proportion of specimens with yield strength above 50 ksi? Calculate the probability using a z-score and the standard normal table or in R. Show work or give code.
d) What is the median yield strength in ksi?
e) What value represents the 25th percentile for A36 grade steel? Show work or give code.