The interval estimate of the mean value of y, given a specific value of x, is commonly represented as a confidence interval. A confidence interval provides a range of values that is likely to contain the true mean of the dependent variable (y) for a specified value of the independent variable (x).
To calculate this interval, we often rely on the results from linear regression analysis. In this context, we first establish a regression line that models the relationship between x and y. For any predicted value of y at a given x, we can compute the standard error of the estimate, which reflects the variability of the data points around the regression line.
Once we have the standard error, the confidence interval can be constructed using the formula:
CI = 1 2 3 4( ext{t-value}) imes ext{SE}Here, B1 is the predicted mean value of y, and SE is the standard error of the prediction. The t-value comes from the t-distribution, and its selection depends on the desired level of confidence (e.g., 95%). Generally, as the confidence level increases, the width of the interval also increases.
This confidence interval thus offers valuable insights, helping researchers and analysts determine the range in which the true mean of y is likely to fall for that specific x value. It is crucial for making informed decisions based on the data and understanding the precision of the estimate.