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The null hypothesis, often denoted as \(H_0\), is a statement that suggests there is no effect or no difference between certain parameters. In the context of hypothesis testing, it serves as the default claim that is tested against observational data.
In this exercise, our null hypothesis states that the population proportions for Treatment A and Treatment B are equal. Formally, this can be written as:

\[ H_0: p_1 - p_2 = 0 \]

Here, \(p_1\) and \(p_2\) are the proportions of success for Treatment A and Treatment B, respectively. The null hypothesis assumes that there's no significant difference between the proportions of success in both treatments.
This hypothesis acts like a starting point. Any observed effect must be strong enough to reject this assumption. If not, we accept the null hypothesis.

The alternative hypothesis, denoted as \(H_1\) or \(H_a\), represents the claim we want to test against the null hypothesis. It indicates the presence of an effect or a difference.

In the given exercise, the alternative hypothesis states that there is a difference between the population proportions for Treatment A and Treatment B. Formally, it can be written as:

\[ H_1: p_1 - p_2 eq 0 \]
The alternative hypothesis suggests that the proportions of success for the two treatments are not equal. This means there is a significant difference in effectiveness between Treatment A and Treatment B.

If the data provides enough evidence to reject the null hypothesis, we accept the alternative hypothesis. This way, the alternative hypothesis represents the new conclusion supported by our test.

The test statistic is a standardized value computed from sample data. It summarizes the data into a single number that we can use to compare against a distribution to make decisions about the hypotheses.

In our exercise, we use the z-test for two proportions. The formula for calculating the test statistic \(z\) is:

\[ z = \frac{(p_1 - p_2)}{\text{Standard Error}} \]

First, we calculate the individual proportions for Treatment A and Treatment B:

\[ p_1 = \frac{84}{(84+21)} \approx\ 0.8 \]
\[ p_2 = \frac{11}{(11+37)} \approx\ 0.229 \]

Next, determine the combined proportion:

\[ p = \frac{(84 + 11)}{(84 + 21 + 11 + 37)} \approx 0.6217 \]
and its complement:
\[ q = 1 - p = 0.3783 \]
Finally, calculate the standard error:

\[ SE = \sqrt{\frac{pq}{n_1} + \frac{pq}{n_2}} = \sqrt{\frac{0.6217 * 0.3783}{105} + \frac{0.6217 * 0.3783}{48}} \approx 0.0577 \]

Putting it all together, we get the test statistic:

\[ z = \frac{(0.8 - 0.229)}{0.0577} \approx 9.89 \]
This value is used to determine the likelihood of observing the data if the null hypothesis is true.

The critical value is a threshold we set based on the significance level \(\alpha\). This value helps determine whether our calculated test statistic falls into the rejection region of the null hypothesis.

For our significance level \(\alpha = 0.05\) and a two-tailed test, the critical values are determined using the standard normal distribution table.

The critical values for \(\alpha = 0.05\) (two-tailed) are:

\[ \pm 1.96 \]
If the calculated test statistic falls beyond this range (i.e., is less than -1.96 or greater than 1.96), we reject the null hypothesis.

In this example, our test statistic \((z = 9.89)\) is much greater than 1.96. Therefore, our test statistic clearly falls in the rejection region, suggesting a significant deviation that leads us to reject the null hypothesis.

The P-value is the probability that the observed test statistic (or one more extreme) would occur if the null hypothesis were true.

In our exercise, we use the z-value obtained to find the P-value using z-tables or statistical software. Given the test statistic \(z = 9.89\), the P-value is incredibly small, essentially zero.

In hypothesis testing, if the P-value is less than the significance level \(\alpha=0.05\), we reject the null hypothesis. Here, the P-value <<< 0.05, indicating strong evidence against the null hypothesis.

This evidence leads us to conclude that there is a significant difference between the population proportions of successes for Treatment A and Treatment B.