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Survey sampling is a vital process in statistics, where a subset of individuals, known as a sample, is selected from a larger group, referred to as a population. The goal is to gather data from the sample to make inferences about the entire population. In our exercise, the sample consists of 2255 randomly selected US adults. It's important that the sample is random to ensure that every individual has an equal chance of being selected, which helps in producing reliable and unbiased results. Random sampling is crucial for representativeness, meaning the sample accurately reflects the diversity of the population, allowing for the generalization of results to a wider populace.

When interpreting survey results, one should consider the sample size and the selection method. A larger sample size typically yields more accurate representations of the population, while a small size can lead to larger errors. This also affects the width of the confidence interval, with larger samples generally leading to narrower intervals, suggesting a more precise estimate of the population parameter.

Proportion calculation involves determining the fraction of the sample that displays a characteristic of interest. It is calculated by dividing the number of individuals with the characteristic by the total number of individuals in the group. In our exercise, there are three proportions calculated:

• Proportion of US adults who use the Internet regularly: \(p_a = \frac{1787}{2255} \approx 0.792\).
• Proportion of US adult Internet users who use a social networking site: \(p_b = \frac{1054}{1787} \approx 0.590\).
• Proportion of all US adults who use a social networking site: \(p_c = \frac{1054}{2255} \approx 0.468\).

These proportions are crucial as they provide a basic measure assessing the prevalence of a characteristic within the sample, which we can then use to make estimations about the population parameters.

The confidence interval formula is used to establish a range of values within which we expect a population parameter to fall, with a certain degree of confidence. The formula for a confidence interval around a proportion is given by:
\[Confidence \, Interval = p \pm Z^* \sqrt{\frac{p(1-p)}{n}} \]
where \(p\) is the sample proportion, \(Z^*\) is the z-score corresponding to the desired confidence level, and \(n\) is the sample size. The z-score represents how many standard deviations away from the mean a particular proportion is within a normal distribution. For a 95% confidence interval, the \(Z^*\) value is approximately 1.96, reflecting the standard score for the 97.5th percentile of a standard normal distribution (as it's one-sided). The result gives us an interval where we expect the true population proportion to lie. In our exercise, the confidence intervals provide ranges for the different proportions of US adults engaged in internet and social networking site usage.

Statistical significance is a determination of whether the results of a study are likely due to a specific factor or simply by chance. It is often assessed using a p-value, which measures the probability of obtaining a result as extreme as the one observed, assuming that the null hypothesis (which typically posits no effect or difference) is true. In the context of confidence intervals, statistical significance is related to whether a confidence interval contains the value of interest or not. For example, if we are determining whether a proportion is significantly different from 50%, we would check if the confidence interval includes 0.5. If it does not, as in case (c) of our exercise (where the confidence interval for the proportion of all US adults who use a social networking site does not include 50%), we would have evidence that the true proportion is statistically significantly different from 50% at the 95% confidence level. Thus, statistical significance in confidence intervals helps us decide whether observed differences in proportions are meaningful or consistent with random variation.