Problem 16 - An Introduction to Mathematical Statistics and Its Applications

Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) and \(Y_{1}, Y_{2}, \ldots, Y_{m}\) be independent random samples drawn from normal distributions with means \(\mu_{X}\) and \(\mu_{Y}\), respectively, and with the same known variance \(\sigma^{2}\). Use the generalized likelihood ratio criterion to derive a test procedure for choosing between \(H_{0}: \mu_{X}=\mu_{Y}\) and \(H_{1}: \mu_{X} \neq \mu_{Y}\).

Step-by-Step Solution

Answer

The decision process for the hypothesis test is based on the computed value of the test statistic and its comparison to the critical values of a standard normal distribution. If the test statistic is within the critical region, it would imply the rejection of \(H_0\) in favor of \(H_1\). Otherwise, \(H_0\) would not be rejected.

Step 1: Computation of sample means

Begin by calculating the sample means, denoted as \(\bar{X}\) and \(\bar{Y}\), for the given samples from normal distributions \(X_{1}, X_{2}, ..., X_{n}\) and \(Y_{1}, Y_{2}, ..., Y_{m}\).

Step 2: Formulation of test statistic

Once the sample means are calculated, create the test statistic, \(\frac{\bar{X} - \bar{Y}}{\sigma*\sqrt{\frac{1}{n} + \frac{1}{m}}}\). This statistic follows a standard normal distribution under \(H_0\), as it is standardized by its standard error.

Step 3: Implementation of generalized likelihood ratio criterion

Use the generalized likelihood ratio criterion to take a decision. This criterion states that reject \(H_0\) if the likelihood ratio \(\Lambda\) is smaller than some threshold. The likelihood ratio is defined as the ratio of the maximum value of the likelihood function under \(H_0\) to the maximum value of the likelihood function under \(H_1\). As such, we will reject \(H_0\) (i.e., decide in favor of \(H_1\)) if \(\Lambda\) is small; that is, if the observed data are much more likely under \(H_1\) than \(H_0\).

Step 4: Decision Making

If the computed value of the test statistic falls in the critical region of the standard normal distribution, then conclude that the null hypothesis \(H_0\) should be rejected in favor of the alternative hypothesis \(H_1\). Otherwise, do not reject the null hypothesis.

Key Concepts

Generalized Likelihood Ratio Criterion

The generalized likelihood ratio criterion is a foundational concept used in hypothesis tests within statistics. In essence, the likelihood ratio measures how well our model with a specific set of parameters aligns with our observed data, compared to an alternative model.

When applying this criterion, we calculate the likelihood ratio, \( \Lambda \), which is the ratio of two likelihoods: the likelihood of the observed data under the null hypothesis (without the effect we're testing for) and the likelihood under the alternative hypothesis (with the effect). The likelihood is the probability of obtaining the observed data given a particular set of parameters. If \( \Lambda \) is too low, it suggests that the data are less likely to occur under the null hypothesis than the alternative, leading us to reject the null hypothesis.

The decision rule involves comparing \( \Lambda \) to a threshold value from a chi-squared distribution. If \( \Lambda \) is smaller than this threshold, we have significant evidence against the null hypothesis. This process allows us to conclude whether or not to reject the null hypothesis using a specific significance level, which is usually denoted as \( \alpha \).

Normal Distribution

The normal distribution is one of the most important probability distributions in statistics, often referred to as the Gaussian distribution. It is symmetrical and bell-shaped, centered around the mean, \( \mu \), with its spread determined by the standard deviation, \( \sigma \).

The ubiquity of the normal distribution stems from the Central Limit Theorem, which states that the sum (or average) of a large number of independent and identically distributed random variables will approximately follow a normal distribution, regardless of the original distribution of the variables.

In the context of hypothesis testing, when we assume the populations are normally distributed, the sampling distribution of the test statistic (like the difference in means) will also follow a normal distribution. This assumption allows us to use standard normal distribution tables to find critical values and probabilities required to make decisions about our hypotheses.

Test Statistic

A test statistic is a numerical value that we calculate from our sample data during hypothesis testing. Its purpose is to quantify the degree to which the observed data deviate from what is expected under the null hypothesis.

The particular test statistic used depends on the type of test being performed and the distribution of the data. In the context of two independent samples from a normal distribution with a known variance, as in our exercise, we use the difference of the sample means standardized by the standard error to yield a test statistic that follows a standard normal distribution under the null hypothesis, given by the formula: \( \frac{\bar{X} - \bar{Y}}{\sigma*\sqrt{\frac{1}{n} + \frac{1}{m}}} \).

This test statistic enables us to make a decision regarding the hypotheses: if the value of the statistic falls into the critical region of the standard normal distribution, we reject the null hypothesis. The critical region is determined based on the chosen significance level \( \alpha \).

Null and Alternative Hypotheses

The null (\(H_0\)) and alternative (\(H_1\)) hypotheses are the backbone of hypothesis testing. The null hypothesis is a statement of 'no effect' or 'no difference,' which serves as a starting point and a claim that is initially presumed to be true. In our example, the null hypothesis is \(H_0: \mu_{X} = \mu_{Y}\), indicating no difference in population means.

The alternative hypothesis is a statement that contradicts the null hypothesis and represents what we are trying to find evidence for. It can be one-sided (specifying a direction of the effect) or two-sided (if we're interested in any difference, regardless of direction). In the exercise, the alternative hypothesis is \(H_1: \mu_{X} eq \mu_{Y}\), suggesting that there is a difference between the two means.

Hypothesis testing involves making a decision about which hypothesis is supported by the sample data. If the evidence (measured by the test statistic) is strong enough to reject the null hypothesis, we accept the alternative hypothesis, bearing in mind that such a decision is subject to error probabilities.