Let Y1, Y2, …, Y20 be a random sample of size n=20 from a...

Let $Y_{1}, Y_{2}, \ldots, Y_{20}$ be a random sample of size $n=20$ from a normal distribution with unknown mean $\mu$ and known variance $\sigma^{2}=5 .$ We wish to test $H_{0}: \mu=7$ versus $H_{a}: \mu>7$ a. Find the uniformly most powerful test with significance level. $05 .$ b. For the test in part (a), find the power at each of the following alternative values for $\mu$ : $\mu_{\alpha}=7.5,8.0,8.5,$ and9.0. c. Sketch a graph of the power function.

Let $Y_{1}, Y_{2}, \ldots, Y_{20}$ be a random sample of size $n=20$ from a normal distribution with unknown mean $\mu$ and known variance $\sigma^{2}=5 .$ We wish to test $H_{0}: \mu=7$ versus $H_{a}: \mu>7$
a. Find the uniformly most powerful test with significance level. $05 .$
b. For the test in part (a), find the power at each of the following alternative values for $\mu$ :
$\mu_{\alpha}=7.5,8.0,8.5,$ and9.0.
c. Sketch a graph of the power function.

Key Concepts

Power of a Test

The power of a test is the probability that the test correctly rejects the null hypothesis when the alternative hypothesis is true. It measures the test’s sensitivity to detect an effect or difference when one truly exists. The power function of a test provides the probability of rejecting the null hypothesis for various values of the parameter under the alternative, allowing for an assessment of the test's performance across different scenarios.

Test Statistic

In the context of a normal distribution with a known variance, the test statistic is typically a standardized form of the sample mean. This statistic is computed by subtracting the hypothesized mean from the sample mean and dividing the result by the standard error. The resulting value follows a standard normal distribution under the null hypothesis, which is critical for determining the rejection region based on quantiles of the normal distribution.

Significance Level

The significance level, often denoted by alpha, represents the probability of committing a Type I error—rejecting the null hypothesis when it is actually true. It is a predetermined threshold that sets the criterion for whether the test statistic is extreme enough to reject the null hypothesis, guiding the formation of the test's rejection region.

Uniformly Most Powerful (UMP) Test

A Uniformly Most Powerful (UMP) test is a statistical test that has the highest power among all possible tests of a given size for every possible value of the parameter under the alternative hypothesis. In tests involving normal distributions, when testing a one-sided hypothesis about the mean and the variance is known, the UMP test is derived from the Neyman-Pearson lemma, leading to a test statistic with a clearly defined rejection region.

Hypothesis Testing

Hypothesis testing is a method in statistics used to decide whether there is enough evidence to reject a presumed null hypothesis in favor of an alternative hypothesis. It involves formulating a null hypothesis (typically a statement of no effect or status quo) and an alternative hypothesis, and then using sample data and probability theory to decide whether to reject the null hypothesis at a predetermined level of significance.

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