An automobile model is known to sustain no visible damage 25

An automobile model is known to sustain no visible damage 25% of the time in 10-mph crash tests. A modified bumper design has been proposed in an effort to increase this percentage. Let p denote the proportion of all 10-mph crashes with this new bumper that result in no visible damage. The hypotheses to be tested are (no improvement) versus The test will be based on an experiment involving n = 20 independent crashes with prototypes of the new design. The natural test statistic here is X = the number of crashes with no visible damage. If H_0 is true, Intuition suggests that an observed value x much larger than this would provide strong evidence against H_0 and in support of H_a. Consider using a significance level of 0.10. The P-value is P(X > x when X has a binomial distribution with That is, the probability of a type I error is just the significance level a. If the null hypothesis is true here and the test procedure is used over and over again, each time in conjunction with a group of 20 crashes, in the long run the null hypothesis will be incorrectly rejected in favor of the alternative hypothesis about 10% of the time. So our test procedure offers reasonably good protection against committing a type I error. There is only one type I error probability because there is only one value of the parameter for which Hq is true (this is one benefit of simplifying the null hypothesis to a claim of equality). Let p denote the probability of committing a type II error. Unfortunately there is not a single value of beta, because there are a multitude of ways for H0 to be false - it could be false because p = 0.30, because p = 0.37, because p = 0.5, and so on. There is in fact a different value of p for each different value of p that exceeds 0.25. At the chosen significance level 0.10, Hq will be rejected if and only if X > 8, so Hq will not be rejected if and only if X

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