A simple random sample of 400 individuals provides 100 Yes responses. a. What is the point estimate of the proportion of the population that would provide Yes responses? b. What is your estimate of the standard error of the proportion, \(\sigma_{\vec{p}} ?\) c. Compute the \(95 \%\) confidence interval for the population proportion.

When statisticians want to estimate characteristics of a large population, they often use a simple random sample (SRS). This is a foundational method to obtain a representative group of individuals from a population where each member is equally likely to be chosen. An SRS allows for the assumption that the sample's properties can be generalized to the whole population.

Take our textbook exercise example: out of a population, we draw a simple random sample of 400 individuals. This means every individual had an equal chance of being selected, and such randomness is crucial to avoid biases that might skew our estimation or confidence intervals. It's a fair way of giving everyone a chance to be included in the sample. Additionally, the size of the sample (400 in this case) needs to be large enough to provide a reliable estimate of the population proportion.

A confidence interval is a range of values that is believed to contain the true population parameter with a specified level of certainty. It adds precision to our point estimate by providing a margin of error. For example, in our exercise, we calculated a 95% confidence interval for the proportion of 'Yes' responses in the population.

The phrase '95% confident' doesn't mean that there's a 95% chance the true proportion lies within our interval. Instead, it implies that if we were to take many samples and build an interval in the same way from each sample, about 95% of those intervals would contain the true population proportion. The actual interval from our one sample might or might not contain it, but we have no way of knowing. It's a statement of reliability or precision, not a probability.

The standard error (SE) measures the variability of a sample statistic, like the sample proportion, from one sample to another. It's based on the standard deviation of the population and the size of the sample. In simpler terms, it indicates how far off we might expect our sample estimate to be from the true population value on average.

In our exercise, we're looking at the standard error of the sample proportion. Since we usually don't know the population's standard deviation, we estimate the standard error using our sample proportion. This tells us about the 'average' error we'd expect in our sample proportion if we repeated our sampling process. It's crucial for constructing confidence intervals, as seen when we multiplied the standard error by a z-score to get the range of our interval. The larger our sample size, the smaller (and thus better) our standard error will be, reflecting a more precise sample estimate.