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The point estimate is a way to estimate an unknown population parameter by using a single value derived from a sample. In the case of population proportion estimation, the point estimate is represented by the sample proportion \( \hat{p} \). This essentially is the best single guess we have based on the sample data.

To calculate \( \hat{p} \), take the number of successes in the sample, which is the number of 'Yes' responses, and divide it by the total number of observations in the sample. For example, if 100 people out of 400 respond 'Yes', the point estimate would be \( \hat{p} = \frac{100}{400} = 0.25 \).

This 0.25 proportion reflects our best single guess of the proportion of the entire population that would say 'Yes.' It is important to remember that this is a point estimate, meaning it does not express the range of uncertainty—just one possible outcome.

The standard error of a statistic provides an estimate of the variability of the sample proportion. It tells us how much the sample proportion \( \hat{p} \) is likely to vary from one sample to another due to chance alone. In simple terms, it allows us to understand how precise our estimate of the population proportion is.

To find the standard error of the sample proportion \( \sigma_{\bar{p}} \), you use the formula:\[\sigma_{\bar{p}} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]where \( \hat{p} \) is the sample proportion, and \( n \) is the sample size.

Using our example: \( \hat{p} = 0.25 \) and \( n = 400 \), thus:\[\sigma_{\bar{p}} = \sqrt{\frac{0.25 \times 0.75}{400}} \approx 0.02165\]
This means there's a standard deviation of 0.02165 between sample proportions, offering a measure of the sampling error, or the natural variation we'd expect just by chance.

A confidence interval gives a range within which we expect the true population proportion to fall. It is constructed around the point estimate and extends as far as the standard error allows, given a particular level of confidence—typically 95%.

For a 95% confidence interval, we use a z-score, which is 1.96 for 95% confidence level. The confidence interval is calculated with:\[\hat{p} \pm z \times \sigma_{\bar{p}}\]Given our point estimate \( \hat{p} = 0.25 \) and standard error \( \sigma_{\bar{p}} \approx 0.02165 \), we compute:\[0.25 \pm 1.96 \times 0.02165 \approx 0.25 \pm 0.04242\]which gives us the range \((0.20758, 0.29242)\).

Thus, we are 95% confident that the true population proportion of 'Yes' responses falls between 0.20758 and 0.29242. The interval reflects the degree of uncertainty or certainty in our estimation, factoring in sample size and variability.