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In hypothesis testing, sample proportions help us understand the characteristics of part of a population. Imagine you are conducting a study on medical treatments, as in the exercise. You are looking at two groups: patients receiving Treatment A and those receiving Treatment B. To know how effective these treatments are, we calculate the sample proportions. For Treatment A, the sample proportion refers to the proportion of patients who survived out of the total number receiving Treatment A. It's calculated as \( p_A = \frac{63}{82} \). This gives us a proportion that represents the success rate of Treatment A. Similarly, for Treatment B, the sample proportion \( p_B \) is calculated as \( p_B = \frac{31}{67} \). This shows the success rate of Treatment B. These proportions provide an initial idea about the effectiveness of each treatment before we dive into further analysis.

The pooled proportion is a crucial concept in hypothesis testing, especially when comparing two proportions, like in the given problem. It provides a way to combine the data from two groups to get a single representation of the overall success across both groups.To calculate the pooled proportion, we add up the number of successes from both groups and divide by the total number of observations in both groups combined. So, for treatments A and B:

• Number of successes for A is 63 and for B is 31.
• The total patients sampled in both groups are 82 and 67, respectively.

The pooled proportion \( p_{pooled} \) is calculated as \( p_{pooled} = \frac{63 + 31}{82 + 67} \).The pooled proportion serves as a baseline to compare individual proportions like \( p_A \) and \( p_B \) against. It plays a significant role in calculating the variability and standard error needed for subsequent testing, allowing us to perform accurate comparisons between treatments.

In statistical tests, particularly hypothesis testing, the normal distribution is often used to decide if observed deviations from expectations are significant. Think of the classic bell curve – that’s what a normal distribution looks like.In our example, after calculating proportions and the pooled proportion, we perform a hypothesis test using the normal distribution. We start by setting up hypotheses:

• The null hypothesis \( H_0 \) assumes no difference, meaning \( p_A = p_B \).
• The alternative hypothesis \( H_A \) suggests Treatment A has a higher survival rate, \( p_A > p_B \).

We then calculate a z-score, which tells us how far and in what direction our sample proportion \( p_A - p_B \) lies from \( 0 \) assuming \( H_0 \) is true. The formula used is \[z = \frac{p_A - p_B}{\sqrt{ p_{pooled} \times (1 - p_{pooled}) \times \left(\frac{1}{n_A} + \frac{1}{n_B}\right) }}\]where \( n_A \) and \( n_B \) are the sample sizes. By comparing the z-score to a standard normal distribution, we find a p-value that helps us decide whether the difference in proportions is statistically significant or not. If the p-value is low, we reject the null hypothesis, supporting that Treatment A is indeed more effective.