A confidence interval is a range of values that is likely to contain the true value of a population parameter with a certain level of confidence. It is used to estimate the precision or accuracy of a sample statistic, such as a mean or a proportion, based on a sample from a larger population.
For example, if we want to estimate the average height of all adults in a certain region, we can take a random sample of individuals from that region and calculate the sample mean height. Then we can construct a confidence interval for the true population mean height, based on the sample mean and the sample size, with a certain level of confidence, such as 95%. This means that if we repeat the sampling process many times, 95% of the resulting intervals will contain the true population mean height.
The formula for a confidence interval is: confidence interval = sample statistic +/- margin of error
The margin of error depends on the sample size, the standard deviation of the population (or the sample, if the population standard deviation is unknown), and the desired level of confidence. For example, if the sample size is larger or the standard deviation is smaller, the margin of error will be smaller, resulting in a narrower confidence interval.
A prediction interval is a range of values that is likely to contain a future observation or outcome with a certain level of confidence. It is used to estimate the uncertainty or variability of a future value based on a statistical model and the observed data.
For example, if we have a regression model that predicts the sales of a product based on its price and advertising budget, we can use a prediction interval to estimate the range of possible sales for a new product with a certain price and advertising budget, with a certain level of confidence, such as 95%. This means that if we repeat the prediction process many times, 95% of the resulting intervals will contain the true sales value.
The formula for a prediction interval is: prediction interval = point estimate +/- margin of error
The point estimate is the predicted value of the outcome variable based on the model and the input variables. The margin of error depends on the residual standard deviation of the model, which measures the variability of the observed data around the predicted values, and the desired level of confidence. For example, if the residual standard deviation is larger or the level of confidence is higher, the margin of error will be larger, resulting in a wider prediction interval. 4
