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When conducting a hypothesis test, the first step is to clearly define the null hypothesis, often denoted as \(H_0\). In this context, the null hypothesis states that there is no difference between two population proportions. It essentially assumes that any observed difference is due to random chance.
For the exercise at hand, \(H_0\) posits that the proportion of people who remove the implants is the same for those born deaf and those who went deaf after learning to speak. Mathematically, this is represented as \(p_1 = p_2\), where \(p_1\) and \(p_2\) are the proportions for each respective group.
The null hypothesis acts as a default assumption, meaning that we maintain it until we have enough evidence to suggest otherwise. Additionally, rejecting the null hypothesis implies that an external factor may be affecting the observed data.

The alternative hypothesis is the counterpart to the null hypothesis, and is symbolized as \(H_a\). It aims to show that there is a statistically significant effect or difference that the null hypothesis fails to predict.
In the given exercise, the alternative hypothesis suggests that the proportion of individuals who remove the implants is different between the two groups. This is mathematically presented as \(p_1 eq p_2\). The alternative hypothesis can be one-sided or two-sided. In this case, it is two-sided, reflecting that either group could have a higher proportion of removals.
The acceptance of the alternative hypothesis over the null hypothesis indicates meaningful differences tied to real-world phenomena, rather than randomness.

The significance level, commonly denoted as \(\alpha\), plays a pivotal role in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is true—commonly referred to as a "Type I error."
For the exercise, \(\alpha\) is set at 0.01, indicating a 1% risk of incorrectly rejecting \(H_0\). The chosen level directly influences the test's rigor: a smaller \(\alpha\) implies greater caution in declaring statistical significance, as more compelling evidence is required to reject the null hypothesis.
Significance levels are typically set before conducting the test to prevent bias. In practice, conventional levels like 0.05 or 0.01 are used, balancing risk tolerance and test practicality.

In hypothesis testing, the p-value serves as a measure of the strength of the evidence against the null hypothesis. A calculated p-value quantifies the probability of observing the test result, or more extreme, given that the null hypothesis is true.
In our scenario, after computing the test statistic \(Z\), the next step is to find the p-value. By checking a standard normal distribution table, you can determine how likely it is to observe a \(Z\) value as extreme as the calculated one, under the assumption that \(H_0\) holds. A two-tailed test is applied here, doubling the p-value from one tail of the distribution.
Concluding the test depends on a comparison between the p-value and the pre-defined significance level \(\alpha\). If the p-value is lower than \(\alpha=0.01\), we reject the null hypothesis, suggesting that the observed difference is statistically significant and unlikely to occur by chance.