A plan for an executive travelers club has been developed by

A plan for an executive travelers\' club has been developed by an
airline on the premise that 13% of its current customers would
qualify for membership, (that is p = 0.13).

Using appropriate probability model(s), ...

Let X denote the number in a random sample of 35 current customers
who qualify for membership. Consider rejecting the company\'s premise
in favor of the claim that p > 0.13 if x >= 8.

What is the probability that the company\'s premise is retained
when it is actually valid (that is 13% qualify)?

What is the probability that the company\'s premise is retained
when in reality p = 0.2 (that is 20% qualify)?

Solution

What is the probability that the company\'s premise is retained
when it is actually valid (that is 13% qualify)?

This is binomial. Number of trials = 35. P = .13. Using a calculator and the binomcdf function,

Binomcdf(35, .13, 7) = .9238.

This returns the probability that 7 or less would qualify. So there is a .9238 probability that 7 or less qualify when p = .13. So the probability that the premise is retained when p = .13 is 0.9238 (answer).

What is the probability that the company\'s premise is retained
when in reality p = 0.2 (that is 20% qualify)?

Changing p to .20, and using the binomcdf function in the calculator:

Binomcdf(35, .20, 7) = .5993.

The probability of getting 7 or less qualifying is now 0.5993 when p = .20. So the probability of retaining the premise (not getting 8 or more) is 0.5993 (answer)