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 \(H_{0}: p_{1}=p_{2}\) vs \(H_{a}: p_{1}>p_{2}\) using \(\hat{p}_{1}-\hat{p}_{2}=0.45-0.30=0.15\) with each of the following sample sizes: (a) \(\hat{p}_{1}=9 / 20=0.45\) and \(\hat{p}_{2}=6 / 20=0.30\) (b) \(\hat{p}_{1}=90 / 200=0.45\) and \(\hat{p}_{2}=60 / 200=0.30\) (c) \(\hat{p}_{1}=900 / 2000=0.45\) and \(\hat{p}_{2}=600 / 2000=0.30\)

Short answer

Without the specific calculated p-values, it is not possible to conclude which sample size provides the strongest evidence for the alternative hypothesis. Generally, with all other factors being equal, a larger sample size would provide stronger evidence for the alternative hypothesis as it minimizes the standard error. However, the comparison of the p-values from all the different sizes will give the correct answer. Use a tool like StatKey, Python, or R to compute the p-values.

Step-by-step solution

Testing Hypothesis for Sample Size (a)

In the first scenario, we have \(\hat{p}_{1}=0.45\) and \(\hat{p}_{2}=0.30\). In this case, the difference for the sample proportion is \(\hat{p}_{1} - \hat{p}_{2}=0.15\). To calculate the p-value, it's recommended to use a technology like StatKey, Python, R, or any kind of statistical software. A p-value less than 0.05 will mean that the result is significant at the 5% level.

Testing Hypothesis for Sample Size (b)

In the second scenario, the sample size is larger with \(\hat{p}_{1}=0.45\) and \(\hat{p}_{2}=0.30\), but the difference \(\hat{p}_{1} - \hat{p}_{2}\) is still 0.15. After calculating the p-value with a statistical tool, like StatKey, if the p-value is less than 0.05, it means that the result is significant at the 5% level.

Testing Hypothesis for Sample Size (c)

In the third scenario, the sample size is now 2,000 with \(\hat{p}_{1}=0.45\) and \(\hat{p}_{2}=0.30\), and the difference \(\hat{p}_{1} - \hat{p}_{2}\) remains 0.15. Again, a statistical tool is used to calculate the p-value. If this p-value is less than 0.05, it's significant at the 5% level.

Comparing the Results

After calculating the p-value for each scenario, the sample size that gives the lowest p-value will provide the strongest evidence for the alternative hypothesis because a smaller p-value means there is stronger evidence against the null hypothesis. You need to compare the p-values of all three tests to find the smallest.

Key concepts

P-value Significance

The p-value is a crucial concept in hypothesis testing that represents the probability of obtaining a sample statistic at least as extreme as the one observed, under the assumption that the null hypothesis is true. In simpler terms, it measures how surprising your data is, given the null hypothesis.

When conducting a hypothesis test, researchers typically set a significance level, often 5%, denoted as \(\alpha = 0.05\), before collecting data. This level represents the threshold for deciding whether an observed effect is statistically significant. If a p-value is smaller than the chosen significance level, the result is deemed significant, which suggests the observed data are unlikely to have occurred by random chance alone, and thus, the null hypothesis is rejected in favor of the alternative hypothesis.

In the exercise above, different sample sizes are used to test the hypothesis. For each, a p-value is calculated using statistical software. If any of the p-values are below 0.05, those results would be considered significant at the 5% level, indicating that there's sufficient evidence to suggest that \(p_1 > p_2\), supporting the alternative hypothesis.

Statistical Software Usage

Statistical software plays an integral role in modern statistical analysis, allowing researchers to manage, analyze, and visualize data more efficiently and accurately. Software such as StatKey, R, and Python have vast libraries and functions designed specifically for performing statistical tests, including hypothesis testing.

Using these tools, researchers can quickly calculate complex statistics, such as p-values, without the need for tedious manual calculations. In the context of the provided exercise, utilizing statistical software simplifies the process of determining whether the sample statistic gives significant evidence to support the alternative hypothesis. This accessibility is vital when dealing with large sample sizes or when precision is paramount for reliable results. By inputting sample proportions and sample sizes into the software, students and professionals alike can effortlessly compute p-values and draw conclusions about their hypotheses.

Sample Size Impact

Sample size is a fundamental factor in statistical tests that directly impacts the reliability of the results. A larger sample size tends to yield more accurate estimates of population parameters and can increase the power of a hypothesis test, which is the probability of correctly rejecting the null hypothesis when it is false. In other words, larger samples reduce the chances of a Type II error, where one fails to detect a real effect or difference.

In the exercise, the impact of sample size is highlighted by comparing three different sample sizes, while keeping the observed differences the same. A larger sample size will typically result in a smaller standard error of the sample statistic, leading to a smaller p-value when testing the same null hypothesis. Therefore, the larger the sample size, the higher the likelihood of detecting the effect size of \(0.15\), assuming such an effect truly exists. It is essential for students to understand that increasing the sample size can be a critical means of providing more convincing evidence for the alternative hypothesis in hypothesis testing.

Null and Alternative Hypothesis

In the realm of hypothesis testing, the null hypothesis \(H_0\) represents a statement of no effect or no difference, and is the hypothesis that one seeks evidence against. On the flip side, the alternative hypothesis \(H_a\) represents the statement for which one is seeking evidence.

In the current exercise, \(H_0: p_1 = p_2\) implies that there is no difference between the two population proportions. Conversely, the alternative hypothesis \(H_a: p_1 > p_2\) suggests that the first population proportion is greater than the second. Hypothesis tests are designed to assess the strength of the evidence against \(H_0\). If evidence is strong enough, as indicated by a low p-value, the null hypothesis is rejected in favor of the alternative. It is crucial to define these hypotheses clearly before conducting the test, as they guide the entire analysis and the interpretation of the results.