Final answer:
The question involves finding the least-squares regression equation based on provided data using a calculator. This equation helps predict the value of a dependent variable y for a given independent variable x. Predictions should only be made within the range of the available data to avoid inaccurate extrapolations.
Step-by-step explanation:
The question is asking for the development of an estimated regression equation that relates a dependent variable y to an independent variable, which is often denoted as x, though the question uses _1 as a placeholder. To find this regression equation, you would first enter the provided data into a graphing calculator or statistical software to create a scatter plot. After creating the scatter plot, you would use the regression function to calculate the least-squares regression line, also known as the line of best fit. This line minimizes the sum of the squared differences between the predicted values and the actual data points.
Once the line of best fit is calculated, the calculator will provide you with an equation in the form y = a + bx, where a is the y-intercept and b is the slope of the line. These values can be rounded to two decimal places for the purpose of making predictions. For example, if the calculator provides you with the regression equation y = -3204 + 1.662x, you can use this to predict y for a given value of x.
If you need to predict y for a specific value of x, such as 2.5 in the given example, you would substitute 2.5 in place of x in the equation and solve for y.
It's important to note when using regression analysis, predictions made outside the range of data provided for x (known as extrapolation) may not be accurate or sensible, such as predicting a negative number of flu cases or making a prediction for a year that is outside the range of the collected data, as illustrated in the example with the year 1970.