A fabrication plant has just completed a contract to supply memory chips to a co

Question

A fabrication plant has just completed a contract to supply memory chips to a computer manufacturer that requires the defective rate of these chips to be no more than 6%. The plant has just installed new equipment to produce these chips. An initial production run of 400 chips will be obtained and one of three actions will be taken depending on the results of this run. If it is shown that more than 6% of the chips are defective, then the equipment will be recalibrated to reduce the defective rate; if it is shown that fewer than 6% of chips are defective, then cheaper raw material will be used to reduce costs; otherwise, the equipment will be unchanged and production will begin. Suppose that among the initial production of400 chips it was found that 28 chips were defective. What action should plant management take at the 5% level of significance? What is the probability that the null hypothesis would be rejected if the population defective rate is actually 8%? Estimate the current defective rate with a 95% confidence interval. In the future plant management would like to estimate the defective rate to within 1% using 95% confidence intervals. What sample size would be required to accomplish this if it is assumed that the defective rate would be no more than 8%?

Explanation / Answer

Solution:

1]

Here, we are given

n = 400

x = 28

P-hat = 28/400 = 0.07

= 0.05

We have to check the following null and alternative hypothesis:

H0: p 0.06 versus Ha: p > 0.06

For checking this hypothesis we have to use the z test for population proportion. The R codes are given as below:

> n = 400

> x = 28

> phat = x/n

> phat

[1] 0.07

> prop.test(x,n,phat)

        1-sample proportions test with continuity correction

data: x out of n, null probability phat

X-squared = 3.393e-32, df = 1, p-value = 1

alternative hypothesis: true p is not equal to 0.07

95 percent confidence interval:

0.04887163 0.09930895

sample estimates:

   p

0.07

Here, we do not reject the null hypothesis that the proportion of defective is greater than 6%.

2]

When defective rate actually 8%

Z = (0.08 – 0.06) / sqrt(0.06*0.94/400) = 1.684304

P(Z<1.684304) = 0.953939

Probability of rejection of null hypothesis = 1 - P(Z<1.684304) = 1 - 0.953939 = 0.046061

3]

95 percent confidence interval:

(0.04887163, 0.09930895) (by using above R-codes)

4]

Now, we have to find sample size

Sample size = n = z^2*p*q/E^2 = 1.96^2*0.08*0.92/0.01^2 = 2827.3137

Required sample size = 2828

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