91Ó°ÊÓ

The sample proportion is a critical concept when it comes to estimating the overall population's characteristics. In the context of surveys or experiments, it represents the fraction of individuals in the sample who possess a certain attribute. Here, the sample proportion is denoted by \( \hat{p} \).
For instance, in the given exercise, we have 1700 respondents out of which 1479 support solar power subsidies. So, we calculate the sample proportion \( \hat{p} \) by dividing the number of favorable responses by the total number of respondents:

• \( \hat{p} = \frac{1479}{1700} = 0.87 \)

This calculation helps provide an estimate of the proportion of the population that might support the subsidies if we looked at everyone, not just those sampled. It's important because it forms the foundation for evaluating other statistical metrics like the confidence interval.

The margin of error (ME) is a measure of the precision of an estimate. It tells us how much a survey result might vary due to sampling randomness. Essentially, it's a buffer around the sample proportion to suggest the range in which the true population proportion is expected to lie with a certain level of confidence.

To determine the margin of error, use the formula:

• \( ME = Z \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \)

For a 95% confidence interval, the critical value \( Z \) is 1.96. So, by substituting the sample proportion value of 0.87 and the sample size of 1700 into this formula, we compute:

• \( ME = 1.96 \cdot \sqrt{\frac{0.87 \times (1 - 0.87)}{1700}} \approx 0.0191 \)

This means the margin of error is approximately 0.0191 or 1.91%. It helps quantify the level of uncertainty inherent in the sampling process.

The population proportion is essentially the goal of our estimation efforts, albeit inferred from the sample data. It represents the fraction of the entire population that exhibits the characteristic in question—in this case, support for solar power subsidies.

While the sample proportion provides a direct estimate, the population proportion aims to generalize this to a broader context. After calculating the sample proportion, statisticians use methods like confidence intervals to approximate the true population proportion. It's essential to differentiate between the estimate from the sample data and the unknown true value from the entire population. Efforts like calculating the confidence interval help us bracket likely values for the population proportion, offering a clearer perspective than singular estimates.

Survey methodology is the backbone of effective data gathering and analysis. It involves a structured approach to collecting and analyzing data, ensuring that the results are representative and accurate. A critical factor in survey methodology is ensuring that the sample is as similar as possible to the entire population it aims to represent.

For the confidence interval to be valid, certain assumptions must be met:

• The sample should be a simple random sample from the population, ensuring each member has an equal chance of being selected.
• The sample size must be large enough to justify using normal approximation, requiring conditions such as \( np \geq 10 \) and \( n(1-p) \geq 10 \).

In our exercise, both conditions are satisfied (e.g., \( np = 1479 \) and \( n(1-p) = 221 \)), signifying that the interval is valid. A robust survey methodology is pivotal in minimizing biases and errors, thereby enhancing the reliability of the findings.