In a survey, the planning value for the population proportion is \(p^{*}=.35 .\) How large a sample should be taken to provide a \(95 \%\) confidence interval with a margin of error of \(.05 ?\)

When conducting a survey, it’s vital to estimate the range in which the true population parameter lies with a degree of certainty. This range is known as the confidence interval (CI). It’s constructed around a sample statistic to include the population parameter—like a population proportion—a certain percentage of the time if we repeated the study multiple times. The 95% confidence interval you often hear about means that if we were to take 100 different samples and compute a CI for each sample, we would expect about 95 of the intervals to contain the population parameter. It's like casting a net in the ocean of uncertainty and hoping to catch the true value most of the time. To broaden or narrow this net, changing the confidence level, consequently, the critical value also changes, affecting the required sample size.

The margin of error is a statistic expressing the amount of random sampling error in a survey's results. It represents the radius of the confidence interval for the population proportion. Think of it as the buffer around your sample's estimate — the smaller you want this buffer to be, the more precise your estimate, but this also requires a larger sample size. The margin of error is influenced by both the size of the sample and the level of confidence desired. In your exercise, wishing to achieve a margin of error of ±.05 while maintaining a 95% confidence interval means you're aiming for a tight buffer, necessitating a fairly substantial sample size to ensure that degree of precision.

The population proportion, denoted by 'p', is a measure that tells us what fraction of the entire population possesses a particular attribute, opinion, or characteristic. In surveys and studies, it is often impractical to measure every individual in a population, so we estimate this proportion by selecting a random sample of the population. In your exercise, the planning value (or assumed true population proportion) is 0.35, signifying that you believe before conducting the survey that 35% of the population has the characteristic of interest. It's a starting point for calculations and can significantly affect sample size calculation — especially when considering variability. Populations with higher variability (proportions near 0.5) require larger samples to accurately estimate the proportion than less variable populations (proportions near 0 or 1).

The idea of statistical significance is at the heart of hypothesis testing and is a measure of how likely it is that an observed outcome, such as a difference between two groups, is due to chance. We use a p-value to quantify this risk. If the p-value is smaller than the pre-established significance level (commonly 0.05), we conclude that the observed effect is statistically significant—meaning that it's unlikely to have occurred due to random chance alone. It's important to note that statistical significance does not necessarily imply practical significance; a result can be statistically significant but may not be large enough to be of real-world importance. In the context of the provided exercise, the significance level relates to how confident we want to be in our estimation of the population proportion; this dictates the critical value which in turn affects how large our sample size should be.