Final answer:
Forecast errors are calculated as the difference between the actual observed value (y) and the predicted value (ŷ). When working with margins of error and sample sizes, error bounds are used to calculate necessary sample sizes or to understand the accuracy of confidence intervals. Error bound formulae are used with known margins of error or for determining the estimate of proportions.
Step-by-step explanation:
Measuring forecast errors is crucial in statistics to evaluate the accuracy of predictions. To compute forecast errors, you subtract the predicted value (ŷ) from the actual observed value (y), resulting in the residual value (y - ŷ). For instance, in regression analysis, if you have a predicted value (ŷ) from a line of best fit and an actual observed value (y), the forecast error would be the difference between these two.
Often, the goal is to minimize these errors to improve the predictive model. In the context of determining the margin of error and sample size, the error bound formula is used to ensure that the margin of error or confidence interval reflects the desired accuracy. For example, with a known margin of error (EBM), researchers can calculate the necessary sample size (n) to achieve a specific confidence level. The error bound for a mean, μ, when the population standard deviation, σ, is known can be found using the appropriate formula.
When working with proportions, the sample proportions (p' and q') are used as estimates for the unknown population proportions (p and q). These estimates are crucial in calculating the error bound in the formula, thereby determining the appropriate sample size or margin of error for a given confidence level.
In summary, the formula for forecast error is Observed y value - predicted y value and when reading statistical studies, error bound or sample mean can often be found using the confidence interval provided.