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Question by kelz: Statistics and Probability….Help Please???
Two different types of injection-molding machines are used to form plastic parts. A part is considered defective if it has excessive shrinkage or is discolored. Two random samples, each of size 300, are selected, and 16 are defective parts are found in the sample from machine 1 while 8 defective parts are found in the sample from machine 2. Is it reasonable to conclude that both machines produce the same fraction of defective parts?

(a) Is it reasonable to conclude that both machines produce the same fraction of defective parts? Use alpha 0.05

(b) Construct a 95% confidence interval on the difference in the two fractions defective.

Best answer:

Answer by Blah
Construct the confidence interval in part b first. If the difference in proportion defective lies outside the interval, then it is NOT reasonable to conclude that they are the same.

To construct the confidence interval:
Calculate the proportion defective and standard deviation (using the binomial formula for stdev) for each machine.

Calculate the difference in proportion defective. Then calculate the standard deviation of the difference by taking the square root of the sum of the individual standard deviations squared. I.e. S_diff = sqrt (s1^2+s2^2).

The Z-value (switching to the normal distribution) corresponding to .475 (half of 95%) is 1.96 (memorize this number — you will use it often). The confidence interval of the difference about a mean of 0 (we’re interested in whether they defect rate is the same — meaning that the difference is zero) is between -1.96*S_diff (rounded to the nearest integer) and +1.96*S_diff (rounded to the nearest integer).

Know better? Leave your own answer in the comments!