Improving on Type II error, Probability of a claim being rejected or accepted using binomial distribution

Question

A manufacturer has developed a new type of bicycle frame which will be sold with a 2-year warranty. To see whether this is economically feasible, 20 prototype frames are subjected to an accelerated life experiment to simulate 2 years of use. The proposed warranty will be modified only if fewer than 90% of such frames would survive the 2-year period. Let p = true proportion of frame survival. Let X be the random variable. Formulate $$H_{0}: p= 0.9; H_{1}:p < 0.9$$ It was found that 14 out of 20 frames did survive.

1. Find Pr(type II error) assuming that true proportion of survival, p= 0.8.
2. Comment on the accuracy of this test. How could you improve this result?

As far as I can see, a type II error will arise if we see x > 15 given p =0.9 (X>15 since $\alpha = 0.05$ since pbinom(X,size=20, prob=0.9) > 0.05 when x >15)

$$\beta = Pr(X > 15; p=0.9) = 1 - Pr(X <=14; p = 0.9) = 1 - pbinom(14, size=20, prob =0.9)= 0.989$$

I am not sure what is meant by the accuracy of this test? $\alpha$? And if so, what would improving the result mean since lowering $\alpha$ would increase $\beta$?

Answer 1

I believe accuracy of test is the power , as power represents the probability of rejecting the false hypothesis. If your null hypothesis is false then the results can be improved by increasing the α , as it represents the Critical Region and Critical Region represents area of rejection, so as you increase area of rejection your probability of rejecting the false null hypothesis increase that leads to increasing the power of a test. Greater the power means less chances of incorrectly accepting the the false null hypothesis(Less chances of committing type II error).