Question

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.

Answer 1

Let x= number of defectives then $X~Binomial (8,p)$

Here p is unknown, we estimate p using method of moment i.e.

$p= \frac{\sum xf}{50}=\frac{17*1+3*2}{50}=0.46$

Hence $P(X = x) = \begin{pmatrix} 8 \\ x \end{pmatrix}(0.46)^x (1 - 0.46)^{8-x} ; x = 0, 1, 2, 3, ..., 8$

$P(X = 0) = (1 — 0.46)^8 = 0.0072\\ P(X = 1) = 8 * (0.46) * (1 — 0.46)^7 = 0.0493\\ P(X = 2) = (8 * 7/2) * (0.46)^2 * (1 — 0.46)^6 = 0.1469\\ P(X > 2) = 1 — P(X = 0) — P(X = 1) — P(X = 2) = 0.7966$

Expected frequencies :

E_o = 50 * P(X = 0) = 50 * 0.0072 = 0.36

E₁ = 50 * P(X = 1) = 50 * 0.0493 = 2.465

E₂ = 50 * P(X = 2) = 50 * 0.1469 = 7.345

E_>2 = 50 * P(X > 2) = 50 * 0.7966 = 39.83