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When conducting a hypothesis test, it all begins with forming the null hypothesis, which is often denoted as \(H_0\). The null hypothesis is a statement of no effect or no difference, and it serves as the default or baseline claim. In the context of our exercise with mummies, the null hypothesis (\(H_0\)) states that the proportion of mummies with artery disease is equal to 40%, just like in modern times. Specifically, we set:

• \(H_0: p = 0.4\)

The null hypothesis presumes that any difference observed between the sample and the population is just due to random chance. Therefore, it is the claim against which evidence is measured.
When performing hypothesis testing, the goal is often to challenge the null hypothesis with evidence suggesting an alternative. This process helps in determining whether observed data is statistically significant enough to warrant a conclusion other than \(H_0\).

Opposite to the null hypothesis is the alternative hypothesis. This is essentially the statement that we want to test, and it is denoted as \(H_A\). Unlike the null hypothesis, the alternative hypothesis suggests the presence of an effect or a difference. In our mummy scenario, the alternative hypothesis Challenges the null by stating:

• \(H_A: p eq 0.4\)

This will propose that the proportion of mummies with artery disease differs from the modern 40%.
A two-tailed test arises from this alternative hypothesis because we are considering the possibility of the proportion being not just greater than but also less than 0.4. In practice, choosing the alternative hypothesis involves specifying the directionality of the test, which determines whether you expect the proportion to differ by being higher, lower, or either direction as specified here.

The right tool for hypothesis testing involving proportions is the Z-test for proportions. This test is particularly useful when comparing a sample proportion to a known population proportion. In our case, the key purpose is to determine whether the proportion (\(\hat{p}\)) of mummies with artery disease from our sample is statistically different from the given population proportion (40%).
The formula used for the Z-test statistic is:

• \(z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}\)

where:

• \(\hat{p}\) is the sample proportion (0.5625 in our case),
• \(p\) is the population proportion under the null hypothesis (0.4), and
• \(n\) is the sample size (16 mummies).

The calculated z-value gives us how many standard deviations the sample proportion is away from the population proportion. This value is then used to find the P-value which helps in deciding whether to reject the null hypothesis.

The significance level is a crucial part of hypothesis testing, denoted typically as \(\alpha\). It represents the probability of rejecting the null hypothesis when it is true, essentially dictating how much risk of error we are willing to accept. In most tests, including our mummy exercise, a common significance level is set at 0.05 or 5%.Setting \(\alpha = 0.05\) means that there is a 5% chance of concluding that the proportion of mummies with artery disease is different from the population proportion even when it is not. It acts as a critical threshold:

• If the P-value calculated from our Z-test is less than \(\alpha\), the result is deemed statistically significant, prompting us to reject \(H_0\).
• If the P-value is above \(\alpha\), there is insufficient evidence to reject the null hypothesis.

This means that choosing the significance level in advance is important as it controls the type I error rate, balancing the hypothesis test between rigor and fairness.