Question

In a process that manufactures bearings, 90% of the bearings meet a thickness specification. A shipment contains 500 bearings. A shipment is acceptable if at least 440 of the 500 bearings meet the specification. Assume that each shipment contains a random sample of bearings. c) What is the probability that between 270 and 280 out of 300 shipments are acceptable? d) What is the probability that 280 out of 300 shipments are acceptable?

Answer 1

c) P(270≤x≤280)=0.572

d) P(x=280)=0.091

Explanation:

The population of bearings have a proportion p=0.90 of satisfactory thickness.

The shipments will be treated as random samples, of size n=500, taken out of the population of bearings.

As the sample size is big, we will model the amount of satisfactory bearings per shipment as a normally distributed variable (if the sample was small, a binomial distirbution would be more precise and appropiate).

The mean of this distribution will be:

`\mu_s=np=500*0.90=450`

The standard deviation will be:

`\sigma_s=√(np(1-p))=√(500*0.90*0.10)=√(45)=6.7`

We can calculate the probability that a shipment is acceptable (at least 440 bearings meet the specification) calculating the z-score for X=440 and then the probability of this z-score:

`z=(x-\mu_s)/\sigma_s=(440-450)/6.7=-10/6.7=-1.49\\\\P(z>-1.49)=0.932`

Now, we have to create a new sampling distribution for the shipments. The size is n=300 and p=0.932.

The mean of this sampling distribution is:

`\mu=np=300*0.932=279.6`

The standard deviation will be:

`\sigma=√(np(1-p))=√(300*0.932*0.068)=√(19)=4.36`

c) The probability that between 270 and 280 out of 300 shipments are acceptable can be calculated with the z-score and using the continuity factor, as this is modeled as a continuos variable:

`P(270\leq x\leq280)=P(269.5<x<280.5)\\\\\\z_1=(x_1-\mu)/\sigma=(269.5-279.6)/4.36=-10/4.36=-2.29\\\\z_2=(x_2-\mu)/\sigma=(280.5-279.6)/4.36=0.9/4.36=0.21\\\\\\P(270\leq x\leq280)=P(-2.29<z<0.21)=P(z<0.21)-P(z<-2.29)\\\\P(270\leq x\leq280)=0.583-0.011=0.572`

d) The probability that 280 out of 300 shipments are acceptable can be calculated using again the continuity factor correction:

`P(X=280)=P(279.5<X<280.5)\\\\\\z_1=(x_1-\mu)/\sigma=(279.5-279.6)/4.36=-0.1/4.36=-0.02\\\\z_2=(x_2-\mu)/\sigma=(280.5-279.6)/4.36=0.9/4.36=0.21\\\\\\P(X=280)=P(-0.02<z<0.21)=P(z<0.21)-P(z<-0.02)\\\\P(X=280)=0.583-0.492=0.091`