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Understanding statistical significance is the cornerstone of hypothesis testing. It's used to determine if the difference or relationship observed in a sample is likely to exist in the broader population. Essentially, it's a measure of how likely it is that an observed outcome is due to chance.

When scientists set a level of significance, commonly denoted as \( \alpha \), they're deciding on the threshold at which they'll reject the null hypothesis, which typically asserts that no effect or no difference exists. If the probability of observing the data, assuming the null hypothesis is true, is less than the significance level, the result is considered statistically significant, and the null hypothesis is rejected.

In the given exercise, the \( \alpha \) levels are set at 0.05, 0.10, and 0.01 for different hypotheses, meaning the researchers have a 5%, 10%, and 1% respective willingness to risk a Type I error, which occurs when the null hypothesis is incorrectly rejected.

The z-test is a statistical test used to determine whether two sample means are different when the variances are known and the sample size is large. It relies on the standard normal distribution and is used in the context of hypothesis testing.

In our exercise, the z-test helps decide whether the sample's proportion (\( p \) in the hypotheses) truly reflects the population proportion. To conduct a z-test, one must calculate the z-score, which is the number of standard deviations a data point is from the mean. We then compare this z-score to a critical value from the z-table, which corresponds to the predefined level of significance.

The critical value in hypothesis testing is a cut-off value that defines the boundary of the rejection region for the null hypothesis. If the test statistic falls into this rejection region, the null hypothesis is rejected in favor of the alternative hypothesis.

In the context of our exercise, the critical values are determined based on the standard normal distribution and the selected significance level \( \alpha \). Different hypotheses use different critical values depending on whether they are one-tailed or two-tailed tests. For example, hypothesis a in the exercise uses a critical value of approximately 1.645, which corresponds to a significance level of 0.05 for a one-tailed (right-tailed) test.

A two-tailed test is used when the alternative hypothesis is not directional, meaning the researcher is interested in detecting any significant difference from the null hypothesis value, regardless of the direction. This requires splitting the significance level \( \alpha \) across two tails of the standard normal distribution.

In our second hypothesis (\(H_{a}: p eq 0.5\)), we employ a two-tailed test. Here, we look for evidence of a significant difference in both the higher and lower ends of the sampling distribution. To reject the null hypothesis, the test statistic needs to be either lower than the critical value in the lower tail or higher than the critical value in the upper tail. In the case of \( \alpha=0.05 \), the critical values are approximately -1.96 and 1.96.

Conversely, a one-tailed test is applied when the direction of the alternative hypothesis is specified. It could be either a lower (left-tailed) or upper (right-tailed) test, depending on whether the expectation is that the population parameter is less than or greater than the hypothesized value.

This test is more powerful for detecting an effect in one direction, as the entire level of significance \( \alpha \) is allocated to one tail of the distribution. For instance, hypothesis a (\( H_{o}: p=0.5 \)) and hypothesis d (\( H_{o}: p=0.7 \)) in the exercise are examples of one-tailed tests—right-tailed to be precise. We look only at the upper end of the distribution for evidence to reject the null hypothesis.