Question

In a manufacturing process where glass products are made, bubbles do occur. When this happens, the quality of the products will be affected.

Suppose based on past records, on average, 1 in every 1000 of these glass products produced has one or more bubbles.

(i) Apply a suitable exact model to compute the probability that a random sample of 5000 glass products will result in fewer than 4 products with bubbles.

(ii) Apply an approximate model to compute the probability that a random sample of 5000 glass products will result in fewer than 4 products with bubbles.

(iii) Comment on the results.

Answer 1

Problem statement:- A manufacturing process is described in the question. Based on this process understanding we have to answer the questions.

Given:- Manufacturing process of glass products is described. Also it is mentioned based on the past experience that 1 in every 1000 glass products suffer from these bubbles. Based on this information we have to answer the questions.

Solution:-

(i):- Based on the information in the problem statement we understand that the most suitable distribution the manufacturing process is following is binomial distribution. with,

probability of success= p= 1/1000=0.001

In the problem statement it is mentioned that sample size =n =5000.

If we observe, the process owner is interested in probability that number of glass products being affected by bubbles is less than 4 =x.

Using the formula,

Probability (Number of glass products<4)=Probability (Number of glass products=0)+Probability (Number of glass products=1)+Probability (Number of glass products=2)+Probability (Number of glass products=3)

=

=0.2648.

(ii):- Approximate model for this binomial distribution is Poisson distribution with Lambda () = n_*p= 5000*(1/1000)=5.

Using the Poisson formula,

Probability (Number of glass products<4)=Probability (Number of glass products=0)+Probability (Number of glass products=1)+Probability (Number of glass products=2)+Probability (Number of glass products=3)

=

= 0.26502.

(iii):- We observe that the exact model and approximate model have almost similar probabilities. The main reason to use approximate model is, it is less computationally intensive. Binomial distribution can be or is approximated to poisson under certain conditions. One condition is when the probability of success (p) is very small (typically less than 0.05. here it is 1/1000). Second condition is number of samples is large (typically greater than 20. here it is 5000). Third condition is when (number of samples)*(probability of success) is less than 10 (here it is 5).

Unde these conditions binomial distribution can be approximated to Poisson distribution. Poisson distribution reduces the computation complexity presented by binomial distribution when sample are large in number and probability of success is too small.