Let ? denote the true average tread life of a certain type of tire, which is ass

Question

Let ? denote the true average tread life of a certain type of tire, which is assumed to be normally distributed
with ? = 1250. Consider testing H0 : ? = 25000 versus Ha : ? > 25000. (a) What decision would you make regarding the null hypothesis in a level .05 test, based on a sample of
size n = 25 with sample mean x ? = 25500? (b) What is the probability of a type II error when the actual value of the mean is ? = 25400.
(c) Find the P -value of the test in part (a) and use it to decide whether to accept or reject the null hypothesis at significance levels ? = .005, ? = .02, ? = .03, ? = .05

Explanation / Answer

The practice of science involves formulating and testing hypotheses, assertions that are capable of being proven false using a test of observed data. The null hypothesis typically corresponds to a general or default position. For example, the null hypothesis might be that there is no relationship between two measured phenomena[1] or that a potential treatment has no effect.[2] The term was originally coined by English geneticist and statistician Ronald Fisher in 1935.[3][4] It is typically paired with a second hypothesis, the alternative hypothesis, which asserts a particular relationship between the phenomena. Jerzy Neyman and Egon Pearson formalized the notion of the alternative. The alternative need not be the logical negation of the null hypothesis; it predicts the results from the experiment if the alternative hypothesis is true. The use of alternative hypotheses was not part of Fisher's formulation, but became standard. It is important to understand that the null hypothesis can never be proven. A set of data can only reject a null hypothesis or fail to reject it. For example, if comparison of two groups (e.g.: treatment, no treatment) reveals no statistically significant difference between the two, it does not mean that there is no difference in reality. It only means that there is not enough evidence to reject the null hypothesis (in other words, the experiment fails to reject the null hypothesis).[5]

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