Final answer:
The margin of error for a poll with 200 respondents and 36% favorability at a 95% confidence level can be calculated using the formula ME = z * √(p*(1-p)/n). As the sample size increases, the margin of error decreases, enhancing the poll's reliability. The margin of error reflects the range within which the true proportion likely falls.
Step-by-step explanation:
To calculate the margin of error for a poll at the 95% confidence level, you need to use the formula for the margin of error (ME) for a proportion, which is ME = z * √(p*(1-p)/n), where z is the z-score corresponding to the 95% confidence level, p is the sample proportion, and n is the sample size. Since 36% of the 200 respondents liked dogs, p would be 0.36, and n would be 200. The z-score for a 95% confidence level is 1.96. Plugging into the formula: ME = 1.96 * √(0.36*(1-0.36)/200), which would result in the margin of error.
It is important to note that as the sample size increases, the margin of error decreases, making the poll more reliable. This principle explains why a larger sample size is desired in public opinion polling and why a typical sample of 1,500 people is likely to have an acceptable margin of error of approximately 2.6 percent. A smaller sample size, on the other hand, like the 9 of 48 respondents in a local poll, will have a greater margin of error, so it's necessary to calculate it correctly for reliable results.
When interpreting the margin of error in the context of a poll, we understand it as the range within which the true proportion of the whole population would fall, with a certain level of confidence. In this case, we are 95 percent confident that the actual percentage of people who like dogs is within the calculated margin of error of the provided figure. This highlights the importance of a low margin of error for precise predictions in polling data.