Question

The same sample statistic is used to test a hypothesis, using different sample sizes. In each case, use StatKey or other technology to find the p-value and indicate whether the results are significant at a \(5 \%\) level. Which sample size provides the strongest evidence for the alternative hypothesis? Testing \(H_{0}: p=0.5\) vs \(H_{a}: p>0.5\) using \(\hat{p}=0.55\) with each of the following sample sizes: (a) \(\hat{p}=55 / 100=0.55\) (b) \(\hat{p}=275 / 500=0.55\) (c) \(\hat{p}=550 / 1000=0.55\)

Short answer

Given the same sample proportion, larger sample sizes provide stronger evidence in favor of the alternative hypothesis, because they lead to larger test statistics and thus smaller p-values. Hence, in this scenario, a sample size of 1000 yields the strongest evidence for the alternative hypothesis.

Step-by-step solution

Understand the given hypothesis

The null hypothesis \(H_0: p=0.5\) states that there is no effect or relationship (here probability is 0.5). The alternative hypothesis \(H_a: p>0.5\) states the existence of an effect or relationship (here probability is greater than 0.5).

Calculate the sample proportion

The sample proportion in each case is given by \(\hat{p} = nr/n\), where \(nr\) denotes the 'number of success cases' and \(n\) is the total sample size. Here, each case gives us the same sample proportion of \(\hat{p}=0.55\).

Compute Test Statistic

Compute the Test statistic \(Z\) using the formula \(Z = (\hat{p} - p)/ \sqrt{(p(1-p)/n)}\). Usually, this would be done using stat software like StatKey or similar as specified in the task. The greater the absolute value of \(Z\), the stronger is the evidence against \(H_0\) in favor of \(H_a\).

Analysis of results

Even though all the sample cases give the same proportion of \(0.55\), the sample size changes. The larger the sample size, the smaller the standard deviation (\(\sqrt{p(1-p)/n}\)), thus leading to a larger test statistic. Consequently, the strength of evidence in favor of the alternative hypothesis increases with the sample size.