Let X1, X2, …, Xn denote a random sample from a normal distribution N(θ, 16). Find the sample si

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Let X1, X2, …, Xn denote a random sample from a normal...

Let $X_{1}, X_{2}, \ldots, X_{n}$ denote a random sample from a normal distribution $N(\theta, 16)$. Find the sample size $n$ and a uniformly most powerful test of $H_{0}: \theta=25$ against $H_{1}: \theta<25$ with power function $\gamma(\theta)$ so that approximately $\gamma(25)=0.10$ and $\gamma(23)=0.90$.

Key Concepts

Uniformly Most Powerful Test

A uniformly most powerful test is one that achieves the highest power among all possible tests of a given size (i.e., significance level) for every parameter value in the alternative hypothesis. This concept is central when trying to decide the best testing procedure in a hypothesis testing problem, particularly when the alternatives are one-sided or specific in nature.

Neyman-Pearson Lemma

The Neyman-Pearson lemma provides a theoretical framework for constructing the most powerful test for a simple null hypothesis versus a simple alternative. Its key insight is that the likelihood ratio statistic is optimal for such hypotheses, and it forms the foundation for developing uniformly most powerful tests under certain conditions.

Power Function

The power function of a test shows the probability that the test correctly rejects the null hypothesis as a function of the true parameter value. It is vital in evaluating the effectiveness of a test, as it highlights the test’s ability to detect alternatives (i.e., its sensitivity) and ensures that the specified power levels are met at particular parameter values.

Sample Size Determination

Sample size determination is the process of calculating the number of observations required to achieve a desired balance between type I and type II errors, often ensuring that the test has sufficient power to detect a meaningful difference from the null hypothesis. This involves solving equations that link the distribution of the test statistic with the desired levels of power and significance.

Hypothesis Testing with Normal Distributions

In the context of normally distributed data with known variance, hypothesis testing typically uses the sample mean as the test statistic, since it follows a normal distribution. This characteristic simplifies the derivation of critical regions and power calculations, making the normal model a common choice for illustrating fundamental concepts in statistical inference.

Type I and Type II Errors

Type I error is the probability of rejecting a true null hypothesis (a false alarm), while type II error is the chance of failing to reject a false null hypothesis. Balancing these errors is essential in designing tests with appropriate characteristics, where controlling the level of the type I error and ensuring sufficient power (1 minus the type II error rate) are key objectives.

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