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In hypothesis testing, setting up the null and alternative hypotheses is a fundamental step. The null hypothesis, often denoted as \( H_0 \), represents a baseline or status quo assumption that we aim to test against. In this scenario, we are examining the claim that at least 70% of investors are confident in meeting their investment objectives. Thus, the null hypothesis is set as \( H_0: p \geq 0.70 \), where \( p \) is the true proportion of confident investors. The alternative hypothesis, denoted as \( H_a \), contains the opposite claim to the null hypothesis and reflects the assertion we want to test for evidence. Here, it is expressed as \( H_a: p < 0.70 \), indicating we are testing whether the actual proportion is less than 70%. A good way to remember this: the null hypothesis often contains an equality or inequality (\( \geq \), \( = \), or \( \leq \)), while the alternative hypothesis reflects a deviation, representing the potential for change or difference.

The p-value is a crucial component in hypothesis testing. It quantifies the strength of the evidence against the null hypothesis. Technically, the p-value is the probability of observing a sample statistic as extreme as the one you have, assuming that the null hypothesis is true. In our example problem, after computing the test statistic using the sample data, we find a p-value of approximately 0.1285. This means there is a 12.85% probability of observing a sample with a proportion of 67% confident investors, or even less, if indeed 70% of investors are confident. When interpreting p-values:

• A small p-value (usually ≤ 0.05) indicates strong evidence against the null hypothesis, prompting its rejection.
• A large p-value (> 0.05) indicates weaker evidence against the null hypothesis, which makes it more reasonable to keep it as is.

In this case, the p-value of 0.1285 is greater than 0.05, providing insufficient evidence to reject the null hypothesis.

The significance level, denoted as \( \alpha \), is the threshold chosen by the researcher to determine when to reject the null hypothesis. It represents the probability of committing a Type I error, which is incorrectly rejecting a true null hypothesis.In many social science research scenarios, the significance level is conventionally set at 0.05 (or 5%). This implies that the researcher is willing to accept a 5% chance of wrongly rejecting the null hypothesis. Applying this to the example problem, the significance level \( \alpha = 0.05 \) sets the decision boundary. Since our computed p-value of 0.1285 is greater than 0.05, we do not reject the null hypothesis. Consider the significance level as a benchmark that helps decide the strength of evidence required in order to believe that an effect exists. While 0.05 is standard, researchers might choose a more stringent level like 0.01 or a more lenient 0.10 based on the context and repercussions of errors.