91Ó°ÊÓ

In the world of statistics, hypothesis testing is a critical method used to make decisions about a population parameter based on sample data.
At its core, hypothesis testing involves two main hypotheses:

• The null hypothesis (\(H_{0}\)) represents the status quo or a statement that there is no effect or no difference.
• The alternative hypothesis (\(H_{1}\)) suggests a possible outcome or effect that we test against the null.

For example, if you're testing whether a new drug is more effective than an existing one, \(H_{0}\) might claim that both drugs have equal effectiveness, while \(H_{1}\) could state that the new drug is more effective.
In the provided exercise, the null hypothesis is \(H_{0}: \theta = \theta^{\prime} = 4\), and the alternative hypothesis is \(H_{1}: \theta = \theta^{\prime \prime} = 9\). The goal is to determine which one these is more likely given the sample data.

The likelihood ratio is a fundamental concept in hypothesis testing, especially within the Sequential Probability Ratio Test (SPRT).
It helps determine how much more likely the data is under one hypothesis compared to another.
In simple terms, the likelihood ratio is the probability of observing the data under the alternative hypothesis divided by the probability under the null hypothesis.In mathematical terms, it is represented as:\[L_n = \left( \frac{\theta^{\prime}}{\theta^{\prime \prime}} \right)^{\frac{n}{2}} \mathrm{exp}\left( \frac{1}{2} \left( \frac{1}{\theta^{\prime}} - \frac{1}{\theta^{\prime \prime}} \right) \sum_{1}^{n} X_{i}^{2} \right)\]This function tells us which hypothesis is favored by the given sample data. In the exercise, this ratio is computed for a series of observations. By comparing it to predetermined thresholds, it helps decide whether to keep collecting data, reject the null hypothesis, or fail to reject it. The key is that if the likelihood under \(H_{1}\) becomes significantly greater, we lean towards rejecting \(H_{0}\).

Decision boundaries are crucial in hypothesis testing as they dictate when and how decisions should be made based on the likelihood ratio.
In the context of the Sequential Probability Ratio Test (SPRT), these boundaries are known as critical thresholds: one for accepting \(H_{1}\) and another for sticking with \(H_{0}\).The boundaries are calculated using:

• If the likelihood ratio, \(L_n\), is greater than the upper boundary, \(c_{1}(n)\), you reject \(H_{0}\) and accept \(H_{1}\).
• If the likelihood ratio is less than the lower boundary, \(c_{0}(n)\), you fail to reject \(H_{0}\).
This setup ensures minimal errors in decision making. For instance, in the exercise, the values \(\alpha_{a}=0.05\) and \(\beta_{a}=0.10\) were used to set:

• The upper boundary: \(c_{1}(n) = 9.5\).
• The lower boundary: \(c_{0}(n) = 0.05556\).
These boundaries help determine whether to continue with the sample collection or make a decision based on the data already gathered.