Statistics 101 — Margin of Error
Proportion of Sample
A country vote for a person, there are 100 people in the country. 57 people vote for A and 43 vote for B, then the 0.43 is the sample mean. The sample variance is
(57*(0- 0.43)² + 43*(1–0.43)²)/(100–1) = 0.2475, s = 0.5
The Sampling Distribution of the Sample Mean
If repeated random samples of a given size n are taken from a population of values for a quantitative variable, where the population means is μ (mu) and the population standard deviation is σ (sigma) then the mean of all sample means (x-bars) is the population mean μ (mu).
As for the spread of all sample means, theory dictates the behavior much more precisely than saying that there is less spread for larger samples. In fact, the standard deviation of all sample means is directly related to the sample size, n as indicated below.
To estimate the SD, we will use our sampling SD as the best estimate for the population SD. Find an interval around the sample mean which I am reasonably confident that there is a 95% chance that the true mean of the population is in that interval.
Example: In a local teaching district, a technology grant is available to teachers in order to install a cluster of four computers in their classrooms. From the 6,250 teachers in the district, 250 were randomly selected and asked if they felt that computers were an essential teaching tool for their classroom. Of those selected, 142 teachers felt that computers were an essential teaching tool.
- Calculate a 99% confidence interval for the proportion of teachers who felt that computers are an essential teaching tool.
- How could the survey be changed to narrow the confidence interval but to maintain the 99% confidence interval?
