91Ó°ÊÓ

The sample proportion is a crucial element in statistical analysis, particularly when estimating population parameters. It provides an estimate of the true proportion within the entire population based on a smaller sample. In the provided exercise, the sample proportion is calculated by dividing the number of teenagers who have made a new friend online (605) by the total number of teens surveyed (1060). This calculation results in a sample proportion of approximately 0.57075, or 57.075%.

Understanding the sample proportion is essential because it serves as the foundation for further calculations, such as the confidence interval. It represents the "best guess" we have regarding the population parameter based on the available sample data. In practical terms, if similar research were conducted multiple times, we would see the sample proportion hover around this number, provided that the sample size and selection methods remain consistent.

The margin of error is an important aspect when reporting a sample statistic like a proportion. It indicates the range within which the true population proportion is expected to lie. In this exercise, the margin of error is determined as part of the confidence interval calculation. Calculating the margin of error involves understanding the variability of the sample proportion and the confidence level you are working with.Here's how it is calculated:

• Identify the desired confidence level, which in this case is 90%.
• Obtain the z-value associated with that confidence level. For a 90% confidence interval, the z-value is 1.645.
• Finally, apply the formula to get the margin of error: \[ME = z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]Using this formula, the exercise yields a margin of error of 0.02025 or 2.025 percentage points.

This value communicates that the true proportion is likely to be within 2.025% above or below our sample proportion of 57.075%. Thus, the margin of error quantifies the uncertainty inherent in the sample estimate of the population proportion.

Statistical interpretation involves making sense of the numbers derived from calculations within a real-world context. When we speak about the 90% confidence interval \(0.5505, 0.591\), we are saying that we have a degree of certainty (90%) that the true proportion of all U.S. teens who have made a new friend online falls within this range. This is not a guarantee but a confident estimate based on the sample data.

Importantly, the entire confidence interval being above 50% provides statistical evidence that more than half of the U.S. teens in the population make new friends online. The interpretation rests upon understanding that confidence intervals offer a range rather than a definitive statement. When analysts say they are "90% confident", they recognize a 10% chance that the population proportion might lie outside this interval.

• This helps in decision-making and policy planning, as we can be reasonably assured of trends in the population based on robust sampling methods and statistical calculation.
• Employing a confidence interval helps in communicating both the estimate and the uncertainty related to the sample proportion.

Ultimately, statistical interpretation links numerical results with the broader understanding of a phenomenon, offering insights that are grounded in empirical evidence.