Question 12: Test the following hypothesis: Ho: the output in the picture below was generated by R code runif(1, 0, 1.5) + runif(1, 0, 1.7). H,: the number was generated by runif(,,) + runif(,,) with some other parameters. The level of significance is a = 0.001. The conclusion is reject the null hypothesis - fail to reject the null hypothesis Console -/ [1] 3.182089 A hint: Let Y be the random variable that is the output of the code and let y, be a sample of size 1. Since y, has a limited precision, we can treat Y as a discrete random variable for the purpose of defining the likelihood ratio (it's easier to speak in terms of probabilities than in terms of probability densities). The denominator of the generalized likelihood ratio doesn't depend on y, as the arguments of runif(,,) + runif(,,) can be adjusted to produce any given number with the probability of 1. The numerator of the likelihood ratio is the probability of the runif(1, 0, 1.5) + runif(1, 0, 1.7) producing y1, it depends on y1. Therefore, if the probability of runif(1, 0, 1.5) + runif(1, 0, 1.7) producing a number bigger than 3.182089 is smaller than o/2 then the likelihood ratio is smaller than 2* and the null hypothesis is rejected. The resulting decision rule is the same as described on page 367 of the text, in the Decision Rules for Nonnormal Data section, but the generalized likelihood ratio approach is more general, it allows for finding a test statistic when, for example, the choice of arguments in the alternative code is restricted in some way. For the purpose of finding the probability of runif(1, 0, 1.5) + runif(1, 0, 1.7) producing a number bigger than 3.182089, it's convenient to treat Y as a continuous random variable