21. textbook 4- 130 An article in The Engineer (Redesign for Suspect Wiring, Jun

Question

21. textbook 4- 130 An article in The Engineer (Redesign for Suspect Wiring, June 1990) reported the results of an investigation into wiring errors on commercial transport aircraft that may produce faulty information to the flight crew. Such a wiring error may have been responsible for the crash of a British Midland Airways aircraft in January 1989 by causing the pilot to shut down the wrong engine. Of 1600 randomly selected aircraft, 8 were found to have wiring errors that could display incorrect information to the flight crew (a) Find a 99% two-sided traditional Cl on the proportion of aircraft that have such wiring errors. (b) Suppose we use the information in this example to provide a preliminary estimate of p. How large a sample would be required to produce an estimate of p that we are 99% confident differs from the true value by at most 0.008? (c) Suppose we did not have a preliminary estimate of p.How large a sample would be required if we wanted to be at least 99% confident that the sample proportion differs from the true proportion by at most 0.008 regardless of the true value of p? (d) Comment on the usefulness of preliminary information in computing the needed sample size.

Explanation / Answer

a)
we know the confidence interval of proportion is given as

p+-z*sqrt(p*(1-p)/n)

here z = 2.58 from the z table for 99% confidence interval

p= 8/1600 = 0.005

so
0.005 +- 2.58sqrt(0.005*(1-0.005)/1600)

solving this

0.00045 and 0.0095

b) These are margin of errors questions

Z0.005*(0.004975/n)1/2 = 0.008 => 2.58*(0.004975/n)1/2 = 0.008

c)

        Z0.01/2*(p*(1-p)/n)1/2 = 0.04 => 2.58*(p*(1-p)/n)1/2 = 0.008

while for p*(1-p), when p=1/2, p*(1-p) arrive maximum and maximum value is ¼, so n =104006*1/4 =26001

d)

When we know the proportion , the sample size needed is really less , as can be seen from the calculation above

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