An airline suspects that a majority of frequent fliers prefer aisle seats. They

Question

An airline suspects that a majority of frequent fliers prefer aisle seats. They define p = proportion of the population of frequent fliers who prefer aisle seats, and plan to test the hypotheses H0: p = 0.5 versus Ha: p > 0.5. The company statistician plans to conduct the test using a significance level of 0.05 based on a random sample of 600 frequent fliers. Suppose in fact 55% of the population of frequent fliers prefer aisle seats. Then the probability that the test will (correctly) result in rejecting the null hypothesis is 0.791. Based on this information, provide numerical values for each of the following.

(a) the null set, 0

(b) the power of the test

(c) the probability of making a Type 1 error (think carefully about this one!)

(d) the probability of making a Type 2 error

Explanation / Answer

b)

The power or sensitivity of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis (H0) when the alternative hypothesis (H1) is true.

As given in the question it is 0.791.

c)

When the null hypothesis is true and you reject it, you make a type I error. The probability of making a type I error is , which is the level of significance you set for your hypothesis test. An of 0.05 indicates that you are willing to accept a 5% chance that you are wrong when you reject the null hypothesis. To lower this risk, you must use a lower value for . However, using a lower value for alpha means that you will be less likely to detect a true difference if one really exists.

Here the probabilit of = 0.05

d)

When the null hypothesis is false and you fail to reject it, you make a type II error. The probability of making a type II error is , which depends on the power of the test. You can decrease your risk of committing a type II error by ensuring your test has enough power. You can do this by ensuring your sample size is large enough to detect a practical difference when one truly exists.

The probability of rejecting the null hypothesis when it is false is equal to 1–. This value is the power of the test.

Typr II probability = 1- = 1-0.791 = 0.209

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