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In hypothesis testing, proportions are an essential metric often needed to understand how two datasets compare. A proportion in statistics represents a part of the whole. It's essentially a percentage that shows how many parts of the data set represent some characteristic. For example, if we consider a sample of voters and find that 55% support a candidate, then this proportion, denoted as \( \hat{p} \), is 0.55.

In our problem, we have two proportions: \( \hat{p}_1 = 0.55 \) and \( \hat{p}_2 = 0.62 \). These represent the proportions of the characteristic of interest in samples 1 and 2. This is crucial for determining if these samples come from populations with the same or different proportions.

• Proportions are calculated by dividing the number of favorable outcomes by the total number of outcomes.
• The differences in proportions are then analyzed to draw conclusions about the populations.

Sample size, denoted as \( n \), is the number of observations in a sample. It is a critical factor in hypothesis testing as it can influence the reliability and validity of the results. Larger sample sizes tend to provide more reliable estimates of population parameters.

In our example, we have two sample sizes: \( n_1 = 300 \) and \( n_2 = 200 \). Each comes from a different population. The larger the sample size, the more accurate the estimate of the population parameter, assuming random sampling.

• A larger sample size reduces the margin of error and increases the confidence in the resulting statistics.
• The sample size affects the calculation of the standard error, which in turn affects the test statistic.

The significance level, often denoted as \( \alpha \), is the threshold at which we decide whether or not to reject the null hypothesis. It represents the probability of committing a Type I error, which is rejecting a true null hypothesis. Common significance levels are 0.05, 0.01, and 0.10.

In this problem, a 1% level of significance is used (\( \alpha = 0.01 \)). This is a strict threshold, indicating that we need strong evidence against the null hypothesis to reject it. The significance level determines the critical value from the Z distribution:

• A smaller \( \alpha \) (like 0.01) means we require stronger evidence against the null hypothesis.
• It corresponds to a confidence level of 99% in this scenario.

The null hypothesis, denoted \( H_0 \), is a statement used in hypothesis testing that assumes no effect or no difference exists. It serves as the default or baseline assumption. In hypothesis tests involving proportions, the null hypothesis usually posits that the two proportions are equal.

In our case, the null hypothesis is \( H_0: p_1 - p_2 = 0 \). This means we assume there is no difference between the two population proportions. The goal of the test is to see if there is enough evidence to reject this hypothesis.

• Starting with the null hypothesis simplifies the testing process by providing a standard to test against.
• The alternative hypothesis (\( H_A \)), in contrast, suggests what we suspect might be true.

The test statistic is a standardized value calculated from sample data during a hypothesis test. It indicates how far the sample statistic is from the null hypothesis if the null is true. In tests of proportions, the test statistic under null hypothesis is often in the form of a Z-score, especially for large samples.

For this exercise, we have calculated the test statistic as \( Z = -2.121 \). This was obtained by taking the difference of proportions and dividing by the standard error:

• The test statistic helps us determine how extreme the sample results are under the assumption that the null hypothesis is true.
• It is then compared to a critical value to decide whether we have enough evidence to reject the null hypothesis.