[DATA] Invest in Education Go to www.pearsonhighered.com/sullivanstats to obtain the data file 12_3_17. The variable “Cost” represents the four-year cost including tuition, supplies, room and board, the variable “Annual ROI” represents the return on investment for graduates of the school—essentially how much you would earn on the investment of attending the school. The variable “Grad Rate” represents the graduation rate of the school.
c. What is the mean annual ROI for a four-year school whose cost is \$180,000?

Verified step by step guidance

1. First, identify the variables involved: the predictor variable is the 'Cost' of the four-year school, and the response variable is the 'Annual ROI'.
2. Next, use the data file 12_3_17 to perform a linear regression analysis where 'Annual ROI' is predicted by 'Cost'. This will give you the regression equation of the form: \(\hat{y} = b_0 + b_1 x\), where \(\hat{y}\) is the predicted Annual ROI, \(b_0\) is the intercept, \(b_1\) is the slope, and \(x\) is the Cost.
3. Calculate the slope \(b_1\) and intercept \(b_0\) using the formulas: \[b_1 = \frac{S_{xy}}{S_{xx}} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}\] \[b_0 = \bar{y} - b_1 \bar{x}\] where \(\bar{x}\) and \(\bar{y}\) are the means of Cost and Annual ROI respectively.
4. Once you have the regression equation, substitute \(x = 180,000\) (the given cost) into the equation to find the predicted mean Annual ROI: \[\hat{y} = b_0 + b_1 \times 180,000\]
5. Interpret the result as the estimated mean annual return on investment for a school with a four-year cost of \$180,000 based on the linear relationship derived from the data.

Key Concepts

Mean (Average)

The mean is a measure of central tendency that represents the average value of a dataset. It is calculated by summing all observations and dividing by the number of observations. In this context, the mean annual ROI indicates the typical return on investment for schools with a given cost.

Return on Investment (ROI)

ROI measures the financial gain or loss generated relative to the cost of an investment. Here, annual ROI reflects how much graduates earn compared to the cost of attending the school, helping evaluate the value of the education investment.

Interpolation and Data Estimation

When exact data points are unavailable, interpolation estimates values within the range of known data. To find the mean annual ROI for a school costing $180,000, one might use nearby data points to estimate the ROI, assuming a relationship between cost and ROI.