A plan for an executive travelers' club has been developed by an airline on the premise that 5% of its current customers would qualify for membership. A random sample of 500 customers yielded 40 who would qualify. a. Using this data, test at level .01 the null hypothesis that the company's premise is correct against the alternative that it is not correct. b. What is the probability that when the test of part (a) is used, the company's premise will be judged correct when in fact 10% of all current customers qualify?

b) The probability that when the test of part (a) is used, the company's premise will be judged correct when in fact 10% of all current customers qualify is < 0.00001

Step-by-step explanation:

Let p be the proportion of customers who would qualify for membership.

[tex]H_{0} :[/tex] p=0.05

[tex]H_{a} :[/tex] p≠0.05

Test statistic can be found using the equation

[tex]z=\frac{p(s)-p}{\sqrt{\frac{p*(1-p)}{N} } }[/tex] where

p(s) is the sample proportion of customers who would qualify for membership ([tex]\frac{40}{500}=0.08[/tex])

p is the proportion assumed under null hypothesis. (0.05)

N is the sample size (500)

Then [tex]z=\frac{0.08-0.05}{\sqrt{\frac{0.05*0.95}{500} } }[/tex] ≈ 3.078

p-value is ≈ 0.002<0.01 therefore we can reject the null hypothesis. There is significant evidence that company's premise is not correct.

b)

To find the probability that the company's premise judged correct where in fact 10% of all current customers qualify, we need to find z-score and p-value of 10% of all current customers qualify under the null hypothesis.

Using above equation zscore of 10% proportion is

[tex]z=\frac{0.10-0.05}{\sqrt{\frac{0.05*0.95}{500} } }[/tex] ≈ 5.13

and P-Value is < 0.00001.