A production engineer monitoring a process that makes magic markers samples 500 markers at random and tests to see how many are defective. He finds 50 defective markers.
(a) Estimate the proportion of defectives in the population.
(b) Construct a 95% confidence interval for the proportion of defectives.
(c) Construct a 99% confidence interval for the proportion of defectives.
(d) How many markers must be tested to form a 99% confidence interval with a margin of error of 1% or less?
(e) How many markers must be tested to form a 90% confidence interval with a margin of error of 1% or less?
(f) The engineer assumes the population proportion of defectives is 10%. Test the hypothesis that the proportion of defectives has risen above 10%, using a = 0.05. Explain what your results mean.
(g) Calculate the p-value for the hypothesis test you just performed, and explain what it indicates about your test. How confident do you feel in your results?
(h) The engineer would like the hypothesis test above to have 95% power for detecting a difference of 1% in the defective rate. How large a sample is needed to achieve this?