New Process, Inc., a large mail-order supplier of women’s fashions, advertises same-day service on every order. Recently, the movement of orders has not gone as planned, and there were a large number of complaints. Bud Owens, director of customer service, has completely redone the method of order handling. The goal is to have fewer than five unfilled orders on hand at the end of 95% of the working days. Frequent checks of the unfilled orders follow a Poisson distribution with a mean of two orders. Has New Process, Inc. lived up to its internal goal? Cite evidence.
Here's one way to do this problem.
Poisson distribution (m = mean):
P(x) = e^(-m) m^x / x!
Note: mean = 2
Find P(0) through P(4). Add together for the probability.
Here's P(0):
P(0) = (e^ -2) (2^0) / (0!)
= (0.1353) (1) / (1)
= 0.1353
I hope this will help get you started.
To determine whether New Process, Inc. has lived up to its internal goal of having fewer than five unfilled orders on hand at the end of 95% of the working days, we need to calculate the probability of having more than five unfilled orders at the end of a day based on the given information.
The Poisson distribution is characterized by a single parameter, λ, which represents the mean or average number of events occurring in a given time period. In this case, the mean number of frequent checks of unfilled orders is two orders.
The probability distribution function of a Poisson distribution is given by:
P(x; λ) = (e^(-λ) * λ^x) / x!
where P(x; λ) is the probability of having x events occur in a given time period with a mean of λ.
To calculate the probability of having more than five unfilled orders, we need to sum the probabilities for x = 6, 7, 8, 9, and so on, up to infinity. Since this is not feasible, we can approximate the probability using a Poisson distribution table or a statistical software.
Assuming we have the necessary probabilities, if the probability of having more than five unfilled orders is less than or equal to 5%, New Process, Inc. would have lived up to its goal of having fewer than five unfilled orders on hand at the end of 95% of the working days.
Without access to the specific probabilities, it is not possible to determine whether New Process, Inc. has met its internal goal.
To determine whether New Process, Inc. has lived up to its internal goal of having fewer than five unfilled orders on hand at the end of 95% of the working days, we can analyze the given information using probability and statistical concepts.
The problem states that the movement of orders has not gone as planned, leading to a large number of complaints. As a result, Bud Owens, the director of customer service, has revamped the method of order handling. The new goal is to have fewer than five unfilled orders on hand at the end of 95% of the working days.
Frequent checks of the unfilled orders follow a Poisson distribution with a mean of two orders. A Poisson distribution is commonly used to model the number of events occurring in a fixed interval of time or space.
To determine if New Process, Inc. has met its goal, we need to find the probability of having more than five unfilled orders at the end of a working day. If this probability is less than or equal to 5% (0.05), then the goal has been achieved.
We can utilize the Poisson distribution to calculate the probability. The probability mass function (PMF) of a Poisson distribution with mean λ is given by:
P(x; λ) = (e^(-λ) * λ^x) / x!
Where:
- x is the number of unfilled orders.
- λ is the average number of unfilled orders (mean), which in this case is 2.
Now, we can calculate the probability of having more than five unfilled orders:
P(X > 5) = 1 - P(X ≤ 5)
To find this value, we need to sum the probabilities for x = 0, 1, 2, 3, 4, and 5.
P(X > 5) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5))
Using the PMF formula, we can calculate each individual probability and then subtract that sum from 1.
By applying the formula, we get:
P(X > 5) = 1 - (e^-2 * (2^0) / 0! + e^-2 * (2^1) / 1! + e^-2 * (2^2) / 2! + e^-2 * (2^3) / 3! + e^-2 * (2^4) / 4! + e^-2 * (2^5) / 5!)
Evaluating this expression will give us the probability of having more than five unfilled orders.
If the resulting probability is less than or equal to 0.05, New Process, Inc. has lived up to its internal goal. Otherwise, the goal has not been met.
Note: To obtain the precise numerical calculation, you can use a scientific calculator or statistical software that supports Poisson distribution calculations.