Question

Let \(\theta\) denote the median of a random variable \(X\). Consider testing $$ H_{0}: \theta=0 \text { versus } H_{1}: \theta>0 . $$ Suppose we have a sample of size \(n=25\). (a) Let \(S(0)\) denote the sign test statistic. Determine the level of the test: reject \(H_{0}\) if \(S(0) \geq 16\) (b) Determine the power of the test in part (a) if \(X\) has \(N(0.5,1)\) distribution. (c) Assuming \(X\) has finite mean \(\mu=\theta\), consider the asymptotic test of rejecting \(H_{0}\) if \(\bar{X} /(\sigma / \sqrt{n}) \geq k\). Assuming that \(\sigma=1\), determine \(k\) so the asymptotic test has the same level as the test in part (a). Then determine the power of this test for the situation in part (b).

Short answer

The level of the test can be determined by using cumulative probabilities of the binomial distribution. The power of the test under the assumption of a Normal Distribution may need simulations for accurate results. Lastly, the threshold (k) for the same level as the original test can be obtained by solving a standard Normal distribution equation. This result then is applied to calculate the power of the asymptotic test.

Step-by-step solution

Determine the level of the test

The level of the test is the probability of rejecting H0 when it's true. Since we are dealing with a sign test statistic, the rejection region is defined as \(S(0) \geq 16\). This corresponds to the probability that a Binomial random variable with n=25 and p=0.5 (under H0) is greater than or equal to 16. Using cumulative probabilities of the binomial distribution, the level of the test can be computed as 1 minus the cumulative probability up to 15: \(1 - BinomialCDF(15; 25,0.5)\)

Determine the power of the test assuming Normal Distribution

Power of a test is the probability of rejecting H0 when H1 is true. In our case, we need to compute the power under H1: X ~ N(0.5, 1). For this, we need to calculate the probability under the distribution of sign statistic under H1, that the sign statistic is greater than or equal 16. This might need simulations to compute if the distribution of the sign statistic under H1 is not easily derived.

Determine k for same level as the original test

We want to adjust the threshold k so that the asymptotic, Z-test with statistic \(\frac{\bar{X}}{\sigma / \sqrt{n}}\) has the same level as the original test. For this, we find the k which yields: \(1 - Phi(k) = level\_original\_test\), where Phi is the standard Normal cumulative distribution function. This means we're equating the right tail of the Z test under H0: N(0,1) to the level of the original test.

Determine the power of the asymptotic test

We want to calculate the power of the asymptotic test when the actual distribution is N(0.5,1). For this, the test statistic under H1: \(\frac{\bar{X}}{\sigma / \sqrt{n}}\) follows a N(0.5, 1) distribution and we compute the right tail beyond k. If Z is the standard Normal, the power is given by: \(1 - Phi(k - 0.5)\)

Key concepts

Hypothesis Testing

Hypothesis testing is a fundamental concept in mathematical statistics used to make inferences about population parameters based on sample data. It often involves formulating two opposing hypotheses: the null hypothesis \(H_0\), which is a statement of no effect or no difference, and the alternative hypothesis \(H_1\), which is what we suspect might be true instead. For example, when testing if a new drug is effective, \(H_0\) might claim the drug has no effect, while \(H_1\) would assert it does.

In hypothesis testing, we use a test statistic to decide whether to reject \(H_0\). This decision is based on how likely it is to observe the data (or something more extreme) if \(H_0\) were true. If this likelihood, known as the p-value, is below a pre-determined significance level \(\alpha\), we reject \(H_0\) in favor of \(H_1\). Importantly, rejecting \(H_0\) doesn't prove \(H_1\) true; it simply provides evidence against \(H_0\) under the assumption that our model of the situation is correct.

Sign Test Statistic

The sign test statistic is used in non-parametric hypothesis testing to determine if there is a statistically significant difference between the median of a population and a specified value. It is particularly useful when the assumptions of parametric tests, such as normal distribution of the data, can't be met. The sign test focuses on the direction of the differences (positive or negative) from the median, ignoring their magnitude.

To calculate \(S(0)\), we would count the number of sample points above the median (\theta) and compare it to a critical value. If \(S(0)\) is greater than or equal to the critical value (as in the textbook example, rejecting \(H_0\) if \(S(0) \geq 16\)), we would reject \(H_0\). This method tests the hypothesis with minimal assumptions and is robust against outliers and skewed data.

Power of a Test

The power of a test is the probability that the test correctly rejects a false null hypothesis \(H_0\). In other words, power represents a test's ability to detect an effect when there is one. A test with higher power is more likely to identify a true effect and less likely to make a Type II error (failing to reject a false \(H_0\)).

Calculating power can be complex, as it depends on the true parameter value, the significance level \(\alpha\), the sample size, and the distribution under the alternative hypothesis \(H_1\). For example, when \(X\) follows a normal distribution with a mean different from 0 (as in part (b)), we can find the power by seeing how much of the distribution falls beyond the critical value used for \(H_0\). High power is desirable in testing, and having adequate power is critical when planning studies to ensure that the test can detect a meaningful difference if it exists.

Asymptotic Test

Asymptotic tests are statistical methods that rely on sample distribution approximations that become accurate as the sample size grows. Instead of working with a finite sample distribution, these tests use theoretical distributions that the sample distribution approaches, that is, the distribution's 'limiting' form.

For large samples, the central limit theorem often justifies the use of the Z-test, an asymptotic test, since the mean of the sample \(\bar{X}\) tends to follow a normal distribution regardless of the original distribution of the data. This allows the computed test statistic, such as \(\bar{X} / (\sigma / \sqrt{n})\), to be compared against a standard normal distribution to make inferences about \(H_0\). In the textbook scenario, the critical value \(k\) is determined to equate the level of the asymptotic test with that of a sign test, illustrating the flexibility and utility of asymptotic methods.

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is denoted by \(N(\mu, \sigma^2)\) where \(\mu\) is the mean and \(\sigma^2\) is the variance.

In practice, the normal distribution is widely used because of the Central Limit Theorem, which states that the average of a large number of independent, identically distributed variables with finite mean and variance tends to follow a normal distribution, regardless of the original distribution. This makes it a pivotal tool in hypothesis testing, especially in tests like t-tests and Z-tests. In the provided exercise, part (b) and part (d) rely on the properties of the normal distribution to compute the sign test's power and to determine the threshold \(k\) for the asymptotic test, respectively.