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The z-statistic is a key concept in hypothesis testing. It represents the number of standard deviations a data point is from the mean. In simple terms, the z-statistic measures how extreme a test statistic is when compared to the null hypothesis assumption. For instance, in our problem, the z-statistic is 2.43. This value tells us how far away our sample proportion is from the hypothesized proportion under the null hypothesis, assuming the data follows a standard normal distribution. The larger the absolute value of the z-statistic, the more evidence there is against the null hypothesis. Understanding the z-statistic involves:

• Calculating how a sample deviates from the population mean under the null hypothesis.
• Interpreting this deviation in terms of standard normal distribution.

The significance level, denoted by \( \alpha \), is a threshold used to decide whether a test result is statistically significant. Common levels are 0.05 and 0.01, which correspond to 5% and 1% respectively. At these levels, we define how confident we are in rejecting the null hypothesis.Choosing a significance level involves balancing the risk of Type I error (rejecting a true null hypothesis). A lower \( \alpha \) means you're more stringent, thus requiring stronger evidence to reject the null. In our exercise, different \( \alpha \) values determine whether our z-statistic falls into the critical region or not, influencing the decision of the test mechanism.

Critical values are the boundaries that separate the critical region from the non-critical region in a distribution. They are crucial in hypothesis testing because they help us decide whether to reject the null hypothesis.For a two-tailed test at \( \alpha = 0.05 \), the critical values are \( \pm 1.96 \), while for \( \alpha = 0.01 \), they are \( \pm 2.576 \). These values are derived from the standard normal distribution.Comparing the test statistic to these critical values determines its significance. If the test statistic falls beyond the critical values, it lies in the critical region, suggesting significant results against the null hypothesis.

A two-tailed test is used when we are interested in deviations in either direction from the hypothesized parameter. In this test, the alternative hypothesis indicates that the population proportion is not equal (either higher or lower) to a specified value.This means that we are considering both extremes in the distribution: too high or too low relative to the mean. Therefore, the critical region encompasses both tails of the distribution. In our case, we have used a two-tailed test with critical values at both \( +z \) and \( -z \), reflecting the possibility of deviations in either direction away from the hypothesized mean. This approach ensures that we catch any significant differences, regardless of their direction.