In statistical analysis, when predicting a dependent variable (y) based on an independent variable (x), we often encounter uncertainty in our predictions. To address this uncertainty, we utilize a prediction interval, which functions similarly to a confidence interval. A prediction interval provides a range within which we expect the actual y value to fall, given a specific x value, with a certain level of confidence—commonly 95%.
To construct a prediction interval, we first need to ensure that the data exhibits a strong linear correlation. This is typically assessed using the correlation coefficient, which should be close to 1 or -1. For instance, if we are analyzing ice cream sales (y) against temperature (x), we would check that our regression line indicates a strong correlation, such as a coefficient of 0.969. Additionally, we must confirm that the x value we are interested in lies within the range of our data set.
Next, we calculate the point estimate (y hat) by substituting the specific x value into the regression equation. For example, if the temperature is 86 degrees Fahrenheit, we would compute y hat using the regression formula, yielding a point estimate of 8,323 for ice cream sales.
Following this, we determine the critical value (t) for our prediction interval, which is derived from the t-distribution based on our desired confidence level and the degrees of freedom (n - 2). For a 95% prediction interval with 7 data points, the degrees of freedom would be 5, leading to a critical value of approximately 2.571.
To quantify the uncertainty in our prediction, we calculate the standard error (s), which can be efficiently obtained using statistical software or calculators. In our example, the standard error is found to be 763.36.
With the point estimate, critical value, and standard error in hand, we can compute the margin of error (E) using the formula:
\[E = t_{\alpha/2} \times s \times \sqrt{1 + \frac{1}{n} + \frac{(x_0 - \bar{x})^2}{n \sigma_x^2 - \sigma_x^2}}\]
Here, \(x_0\) is the specific x value (86), \(\bar{x}\) is the mean of the x values, and \(\sigma_x\) is the standard deviation of the x values. After performing the calculations, we find the margin of error to be 2,268.3.
Finally, we establish the prediction interval by adding and subtracting the margin of error from the point estimate. This results in a lower bound of 6,054.7 and an upper bound of 10,591.3. Thus, we can confidently state that we are 95% certain that when the temperature is 86 degrees Fahrenheit, ice cream sales will fall between these two values.
