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Sample proportions are a way to estimate the proportion of a characteristic in a population using a smaller, more manageable subset called a sample. For instance, if you want to understand how many people prefer tea over coffee in a city, instead of asking everyone, you can take a sample.
Taking a sample involves selecting a group of people and finding the proportion of them who prefer tea. If 165 out of 300 people prefer tea, the sample proportion (\( \hat{p} \)) is 0.55.
In this exercise, we compute sample proportions for two different samples:

• Sample 1 has \( \hat{p}_1 = 0.55 \) with a sample size \( n_1 = 300 \).
• Sample 2 has \( \hat{p}_2 = 0.62 \) with a sample size \( n_2 = 200 \).

These sample proportions help us understand the characteristic or preference of the two distinct groups.
Finally, the difference \( \hat{p}_1 - \hat{p}_2 \) indicates how these proportions compare to each other.

The standard error measures variability. Specifically, it tells us how much we expect the sample proportion to differ from the true population proportion.
It provides insight into how well our sample represents the population.
The key idea behind standard error for two sample proportions is combining the variability from two groups.
This is calculated using the formula: \[ \sqrt{ \frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2} } \]In our context, using values

• \( \hat{p}_1 = 0.55 \), \( n_1 = 300 \), \( \hat{p}_2 = 0.62 \), \( n_2 = 200 \).

This gives a standard error of \( 0.056 \).
The smaller the standard error, the more stable our estimate, helping make better predictions about population differences.

A z-score gauges how far a data point is from the mean, measured in standard deviations.
For confidence intervals, z-scores point to the number of standard deviations a statistic, like a sample proportion, is from the estimated population parameter.
For a given confidence level, a specific z-score corresponds.
For example, a 99% confidence level results in a z-score of approximately \( 2.576 \).
This means we can be 99% confident that the true difference in proportions lies within the range calculated using this z-score.

• Higher confidence levels yield higher z-scores.
• Higher z-scores denote a wider interval.

Keeping this in mind helps us understand what range our difference in sample proportions could have when calculating the interval.

The confidence level indicates how sure we are that the interval contains the true population parameter.
Its common values include 90%, 95%, and 99%.

• A 99% confidence level means we are sure, almost all of the time, that our interval covers the actual difference in proportions between two groups.

The interval is calculated as\[(\hat{p}_1 - \hat{p}_2) \pm Z_{\alpha/2} \times \text{standard error}\]In our case:

• The difference is \(-0.07\).
• Z-score for a 99% level is approximately \(2.576\).
• The standard error is \(0.056\).

So, the interval becomes approximately \(-0.214\) to \(0.074\).
This implies, with 99% confidence, the true difference of proportions lies within this range, helping us make informed decisions about our hypothesis.