Question

Let \(X\) have a binomial distribution with the number of trials \(n=10\) and with \(p\) either \(1 / 4\) or \(1 / 2 .\) The simple hypothesis \(H_{0}: p=\frac{1}{2}\) is rejected, and the alternative simple hypothesis \(H_{1}: p=\frac{1}{4}\) is accepted, if the observed value of \(X_{1}\), a random sample of size 1 , is less than or equal to 3 . Find the significance level and the power of the test.

Short answer

Firstly, calculate the probabilities of \(X_{1}\) being less than or equal to 3, under the conditions set by \(H_{0}\) and \(H_{1}\) respectively. Thus the significance level of the test is calculated using the probability under \(H_{0}\) and power of the test is calculated using the probability under \(H_{1}\).

Step-by-step solution

Understanding the Problem

The problem has a random variable \(X\) which follows a binomial distribution. The number of trials \(n\) is given as 10. The probability of success \(p\) is asked to be assumed under two conditions - either \(1 / 4\) or \(1 / 2\). This is tested under two hypotheses - \(H_{0}\) and \(H_{1}\). The test rejects \(H_{0}: p=\frac{1}{2}\) and accepts \(H_{1}: p=\frac{1}{4}\) if the observed value of \(X_{1}\) (a random sample of size 1) is less than or equal to 3. The task is to calculate the significance level of the hypothesis test under these conditions and also the power of the test.

Define Significance Level

Significance level is defined as the probability that the test rejects the null hypothesis \(H_{0}\) when it is true. So, for a test with significance level \(\alpha\), we will reject \(H_{0}\) if \(X_{1}\) is less than or equal to 3 under the conditions of \(H_{0}\). Thus, \(\alpha = P(X_{1} \leq 3 | p = \frac{1}{2})\). Since \(X_{1}\) follows binomial distribution with \(n = 10\) and \(p = \frac{1}{2}\), we can calculate this probability using cumulative binomial probability formula.

Calculation of Significance Level

The probability \(\alpha = P(X_{1} \leq 3 | p = \frac{1}{2})\) is calculated by adding the probabilities for \(X_{1}\) equals to 0, 1, 2 and 3. Using binomial formula, we get \(\alpha = \sum_{k=0}^{3} C_{10}^{k} \cdot (\frac{1}{2})^{k} \cdot (1 - \frac{1}{2})^{10 - k}\)

Define Test Power

The power of a test is the probability that we reject the null hypothesis \(H_{0}\) when the alternative hypothesis \(H_{1}\) is true. Therefore, the power of this test is calculated as \(\beta = P(X_{1} \leq 3 | p = \frac{1}{4})\), again using the binomial distribution.

Calculation of Test Power

The power \(\beta = P(X_{1} \leq 3 | p = \frac{1}{4})\) is calculated by adding the probabilities for \(X_{1}\) equals to 0, 1, 2 and 3. Using binomial formula, we get \(\beta = \sum_{k=0}^{3} C_{10}^{k} \cdot (\frac{1}{4})^{k} \cdot (1 - \frac{1}{4})^{10 - k}\)