Hypothesis testing is an essential statistical tool used to decide between two competing hypotheses based on sample data. In this exercise, we examine two statistical hypotheses, known as the null hypothesis \(H_0\) and the alternative hypothesis \(H_1\).
- The null hypothesis \(H_0\) assumes that data is uniformly distributed between \(0\) and \(\theta\).
- The alternative hypothesis \(H_1\) suggests that the data follows an exponential distribution with rate \(\frac{1}{\theta}\).
The goal is to determine which hypothesis is more likely given the sample data. A common approach is the likelihood ratio test, which compares the likelihoods of the data under both hypotheses.
The test statistic, the likelihood ratio \(\Lambda\), helps us make this decision. A high or low value (based on a threshold) indicates which hypothesis is more plausible, though setting this threshold often depends on the context and desired confidence level.