07/12

2015

A confidence interval for the population mean gives an indication of how accurately the sample mean estimates the population mean. A 95% confidence interval is defined as an interval calculated in such a way that if a large number of samples were drawn from a population and the interval calculated for each of these samples, 95% of the intervals will contain the true population mean value.

A prediction interval gives an indication of how accurately the sample mean predicts the value of a further observation drawn from the population.

Confidence interval defines like: *estimate ± margin of error*.

A 95% confidence interval (CI) is twice the standard error (also called margin of error) plus or minus the mean. In our example, suppose the *mean* is 990 and *standard deviation* as computed is 47.4, then we would have a confidence interval (895.2, 1084.8) i.e. 990 ± 2 * 47. 4 . If we repeatedly choose many samples, each would have a different confidence interval but statistics tells us 95% of the time, CI will contain the true population *mean*. There are other stringent CIs like 99.7% but 95% is a golden standard for all practical purposes.

The simplest way to obtain a confidence interval for a sample mean is with the t.test function, which provides one with the output.

x = rnorm(30, 195, 17) t.test(rnorm(30, 195, 17)) One Sample t-test data: rnorm(30, 195, 17) t = 59.174, df = 29, p-value < 2.2e-16 alternative hypothesis: true mean is not equal to 0 95 percent confidence interval: 190.1872 203.8047 sample estimates: mean of x 196.9959

From the results, you can see that the mean is 196.99, with a 95% confidence interval of 190.18 to 203.804.