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Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. This process involves forming two hypotheses:

• The Null Hypothesis (\( H_0 \)): a statement that indicates no effect or no difference, suggesting that any observed deviation is due solely to chance.
• The Alternative Hypothesis (\( H_a \)): a statement indicating the presence of an effect or a difference.

The aim is to determine whether the available data reject the null hypothesis in favor of the alternative one.
The hypothesis test is conducted at a specific significance level, often set at 5% or 1%, representing the probability of making a mistake by rejecting \( H_0 \) when it is indeed true. This probability level forms a threshold against which the evidence is evaluated.

Critical values are fundamentally important in hypothesis testing, serving as the demarcation lines set by the significance level. They determine the "cut-off" points that dictate whether a test statistic falls within the acceptance region or the rejection region. For example:

• In a one-sided hypothesis test at a 5% significance level, the critical value could be \(-1.645\) for left-tailed tests.
• At the 1% significance level, it may be \(-2.33\) for left-tailed tests.

These values can vary depending on whether the test is one-sided or two-sided.
For two-sided tests, the critical values at the 5% level are \( 卤1.96 \), and at the 1% level, they are \( 卤2.576 \). Observing a test statistic beyond these critical values indicates a significant result, leading to a rejection of the null hypothesis.

The null hypothesis, denoted as \( H_0 \), is the bedrock assumption of our statistical test. It posits that there is no effect or difference in the population parameter that we are trying to validate. In the provided example, the null hypothesis \( H_0: p = 0.65 \) suggests that the proportion is statistically equal to 0.65.
This hypothesis serves as a starting point for statistical testing. The reason for beginning with \( H_0 \) is to assume as a default stance that the observed effect is simply due to random chance.
The decision in hypothesis testing usually revolves around rejecting or not rejecting \( H_0 \). If the test statistic falls in the rejection region, as determined by the critical values, the alternative hypothesis is supported instead.

The alternative hypothesis, denoted as \( H_a \), is the statement that researchers are usually most interested in proving. It postulates that there is a specific effect or difference worth investigating.
Depending on the test, \( H_a \) can take different forms:

• For a one-sided test: \( H_a: p < 0.65 \) (as in the original problem), meaning we suspect the proportion is less than 0.65.
• For a two-sided test: \( H_a: p eq 0.65 \), meaning any deviation from 0.65鈥攚hether positive or negative鈥攊s of concern.

The confirmation of the alternative hypothesis indicates that the null hypothesis is not likely to be true, implying that the observed effect may indeed exist in the population. A rejection of \( H_0 \) provides support to \( H_a \), thereby highlighting its role as the main focus of interest in most significance tests.