A manufacturing company claims that no more than 5% of the parts made by the com

Question

A manufacturing company claims that no more than 5% of the parts made by the company have some type of defect. A quality control specialist randomly selects 300 parts from a total of 5,000 parts made over a given time period to test the parts for defects. The specialist classifies the individual parts as either having a defect or not having a defect. For the 300 parts selected by the employee, 35 were determined to have some type of defect. Note that the population is more than 10 times the sample size and assuming independence, a binomial distribution can be used to model the probability that X = 0 to X = 300 parts have some type of defect.

1. Determine if the conditions for using a normal distribution to approximate the binomial distribution are satisfied. Check that both np and n(1-p) are greater than 10.

2. Using the claim p= 0.05 , construct a 95% confidence interval, that is based on this claim, for the proportion of defective parts using a normal distribution to approximate to the binomial distribution. Give a correct interpretation of the confidence interval in this context.

3.Based on the confidence interval found and the specialist’s findings, is there evidence that the true proportion of parts that have some type of defect is greater than 5%? Explain.

Explanation / Answer

(1) here p=0.05, n=300, np=300*0.05=15 and n(1-p)=300*(1-0.05)=285,

since both np and n(1-p) is greater than 10, so we approximate to the normal distribution

(2) p=0.05 and SE(p)=sqrt(p(1-p)/n)=sqrt(0.05*(1-0.05)/300))=0.0126

(1-alpha)*100% confidence interval for population proportion (P)=sample proportion (p) ±z(alpha/2)*SE(p)

95% confidence interval for P=p±z(0.05/2)*SE(p)=0.05±1.96*0.0126=0.05±0.0235=(0.0265,0.0735)

(3) true proportion of defective =35/300=0.1167,

since this proprotion=0.1167 does not belong to the interval (0.0265,0.0735) , so there is evidence that the true proportion of parts that have some type of defect is greater than 5%.

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