Question

Please show and answer all the parts of this question.

Suppose that in manufacturing a very sensitive electronic component, a company and its customers have tolerated a 2% defective rate. Recently, however, several customers have been complaining that there seem to be more defectives than in the past. Given that the company has made recent modifications to its manufacturing process, it is wondering if in fact the defective rate has increased from 2%. For quality assurance purposes, you decide to randomly select 1,000 of these electronic components before they are shipped to customers. Of the 1,000 components, you find 25 that are defective. Assume that the company produces a very large number of these components on any given day.

1. Set up an appropriate hypothesis to test whether or not the defect rate has increased.
2. Before proceeding to test your hypothesis, check that all assumptions and conditions are satisfied for such a test.
3. Conduct the test using a .05 level of significance (alpha) and state your decision about whether or not you believe that the defect rate has increased.
4. What would be the minimum number of defectives in a random sample of 1,000 would you need to find in order to statistically decide that the defect rate exceeds .02 (again, assuming a .05 level of significance).            

Answer 1

Given x=25, n=1000, alpha = 0.05

(a) setting up hypothesis

Null hypothesis : defective rate of components is 2%

Alternate hypothesis: defective rate of components has increased from 2%

(b) Since company produces a very large number of these components on any given day, we can assume

, where N is total number of components produced.

np = 1000 * 0.02 = 200 >10

np(1-p)=1000*0.02*(1-0.02) = 19.6 > 10

Hence we can assume shape of sampling distribution of sample proportion is approximately normal.

(c) sampling proportion  

Since it is right tailed test

z-critical value when alpha is 0.05 is = 1.64

p-value (area to the right of z-test = 1.13) = 0.1294

Since or p-value is greater than level of significance ( alpha = 0.05) , we failed to reject the null hypothesis.

This means there is not enough evidence to support the claim that defective components rate has been increased from 2%

(d)

when alpha is 0.05, z-critical = 1.64

z-test statistics should be greater than this value for rejecting the null hypothesis and concluding that defective rate has been increased from 2%

x > 26.888

Rounding to nearest next integer, minimum number of defectives should be 27 in order to statistically decide that the defective rate has been increased.