Question

An airline claims that on average 5% of its flights are delayed each day. On a given day, of 25 flights, 3 are delayed. Set a level of significance at 0.01.

(1) Does the measurement on that day support the hypothesis that true proportion is not 5%? List and test the hypothesis in details.

(2) For the above hypothesis, if the actual proportion is 10%, what is the probability of a type II error β ?

Answer 1

1) The sample proportion here is computed as:

p = x/n = 3/25 = 0.12

The test statistic here is computed as:

As this is a two tailed test, the p-value here is computed from the standard normal tables as:

p = 2P( Z > 1.6059) = 2*0.0541 = 0.1082

Therefore as the p-value here is 0.1082 > 0.01 which is the level of significance, therefore the test is not significant and we cannot reject the null hypothesis here. Therefore we dont have sufficient evidence to reject the claim that the delayed rate is 5%.

b) For 0.01 level of significance, we have here:
P( -2.576 < Z < 2.576 ) = 0.99

For a true proportion of 0.1 that is 10%, the probability of type II error is computed here as:

P( Z > 2.576 )

Therefore the probability here is computed as:

P( p > 0.1623)

Converting it to a standard normal variable, we get here:

getting it from the standard normal tables, we get here:

Therefore the probability of type II error here is given as: 0.1496