In a poll of 100 adults, 45% reported they believed in “faith healing.” (USA Today, 20 April 1998). Based on this survey, a “95% confidence interval” for the proportion in the population who believe in faith healing is about 42% to 48%. If this poll had been based instead on 5000 adults, do you think the “95% confidence interval” would be wider or narrower than the interval given? Explain.
To understand how the "95% confidence interval" would change when the sample size increases from 100 adults to 5000 adults, we need to consider the concept of sampling error and the relationship between sample size and precision of estimates.
In statistics, sampling error refers to the variability observed between different samples taken from the same population. A larger sample size generally reduces sampling error and improves the precision of estimates. The "95% confidence interval" quantifies the range in which the true population parameter (proportion in this case) is likely to exist.
When the sample size increases from 100 to 5000, the confidence interval would become narrower. This is because a larger sample size means more data points are being considered, reducing the uncertainty associated with the estimate.
To determine the magnitude of the change in the confidence interval, we can use the formula for calculating the margin of error, which is based on the sample size and the variability of the data. As the sample size increases, the margin of error decreases, resulting in a narrower confidence interval.
In the given scenario, with a sample size of 100, the estimated proportion of adults who believed in faith healing is 45%, with a "95% confidence interval" of 42% to 48%. But if the sample size were 5000 instead, we would expect the confidence interval to be narrower, meaning it would have a smaller range and be more precise.
In summary, the "95% confidence interval" for the proportion of adults who believe in faith healing would be narrower when based on a sample size of 5000 adults compared to a sample size of 100 adults. This increased precision is due to the larger sample size, resulting in reduced sampling error.