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Hypothesis testing is a fundamental statistical process used to determine if there is significant evidence to reject a null hypothesis (\(H_0\)). In the context of this exercise, the null hypothesis states that the true proportion \(p\) equals 0.5. The alternative hypothesis (\(H_a\)) is what you seek to prove. In part (a) of the exercise, the alternative hypothesis suggests that \(p\) > 0.5, indicating a one-tailed test. Alternatively, in part (b), \(H_a: p eq 0.5\)is explored for a two-tailed test.
The whole idea is to analyze evidence in the form of data to see if the observed outcomes provide enough support to shift from believing the null hypothesis to favor the alternative. Essentially, hypothesis testing helps to make informed decisions about populations based on sample data.

Key steps in hypothesis testing involve:

• Stating the null and alternative hypotheses.
• Choosing a significance level (common levels are 5% and 1%).
• Calculating a test statistic, like a Z-score, based on the sample data.
• Comparing the test statistic to critical values to determine if results are statistically significant.

Critical Z-scores are the points on the Z-distribution that define the boundary or cutoff for the rejection of the null hypothesis. They depend on the significance level and whether the test is one-tailed or two-tailed. Typically, critical Z-scores are determined by using statistical tables or software.
For a one-tailed test with a 5% significance level, the critical Z-score is 1.645. For a 1% level, it is 2.33. These values represent the points such that any observed Z-score greater than these values would lead to rejecting the null hypothesis.
In a two-tailed test, the critical Z-scores are slightly different due to the distribution of the probability area across both tails of the bell curve. Here, for a 5% level, the critical Z-scores would be ±1.96, and for a 1% significance level, ±2.58. These values set the thresholds that an observed Z-score must surpass for the null hypothesis to be rejected.

These cutoff points are crucial because they help prevent incorrect conclusions – such as rejecting the null hypothesis when it's true, known as a Type I error.

A one-tailed test in hypothesis testing is used when the alternative hypothesis predicts a specific direction of the effect. For example, in the given exercise, part (a) uses a one-tailed test with an alternative hypothesis \(H_a: p > 0.5\). This means we're specifically interested in seeing if there is evidence that the proportion is greater than 0.5.
The key advantage of a one-tailed test is that it has more power to detect an effect in one direction, as all the significance level is concentrated in one tail of the distribution. This can make it more powerful for detecting meaningful effects in that direction.
The decision to reject the null hypothesis is based on whether the calculated test statistic (e.g., Z-score) exceeds the critical value on the specified tail. Here, because 2.19 exceeds the critical value of 1.645 at the 5% significance level, it indicates statistical significance, leading to a rejection of the null hypothesis.

A two-tailed test is used when the alternative hypothesis does not predict the direction of the effect but rather suggests any difference, either more or less, from the null hypothesis. In the exercise, part (b) addresses a two-tailed test where \(H_a: p eq 0.5\). This looks for evidence that the true proportion is different from 0.5, regardless of whether it's higher or lower.
Two-tailed tests are more conservative than one-tailed tests because the significance level is split between both tails of the distribution. This means that for the same significance level, a two-tailed test requires more evidence to reject the null hypothesis compared to a one-tailed test.
In this exercise, the Z-score of 2.19 is compared against critical values of ±1.96 for the 5% significance level and ±2.58 for the 1% level. Since 2.19 is greater than 1.96 but less than 2.58, we reject the null hypothesis at 5% but fail to reject it at 1%. This demonstrates the broader applicability of two-tailed tests in capturing any significant deviations from hypothesized values.