"Finding the Coefficient of Determination and the Standard Err...

Question

"Finding the Coefficient of Determination and the Standard Error of Estimate In Exercises 11-20, use the data to (b) find the standard error of estimate s_e and interpret the result.

12. [APPLET] Median and Mean Hourly Wages The table shows the median and mean hourly wages (in dollars) in 10 states in a recent year. The equation of the regression line is y = 1.208x + 1.495. (Source: U.S. Census Bureau)

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Accepted Answer

All right. Hello, everyone. So, this question says, a real estate analyst recorded the average house size in hundreds of square feet and corresponding sale price in thousands for 10 neighborhoods. Find the standard error of estimates C and interpret the result. Here we have 4 different answer choices labeled A through D. All right, so first and foremost, in addition to the data table provided, we also have an equation for the linear regression model. The reason this matters is because this is going to help us determine our standard error of estimate. Now Miss scroll Now S sub e. Is equal to the square root of the sum of all differences between. Why and why hat squared. Divided by and subtracted by 2. Where N is the sample size. Now Why hat in this context represents the expected value obtained from the linear regression model. So, there's additional information that you have to find, or you have to calculate. In addition to each X and Y value that we're already given for each neighborhood. You also have to find A value for white bar. Corresponding to each data point, and you would obtain this by plugging in the corresponding value of X into the linear regression model. Once you've obtained a value for Y bar, you would then take each Y value listed and subtract that by Y bar. Now, the value for Y subtracted by Y bar is referred to the residual. So then after you found each residual, you would then square them. So for each data point, you would then square the residual, the difference between Y and Y hat. Earlier I said why bar, but I actually meant why hat, so I just wanted to clarify. All right, so ultimately, it's this last column of data that is going to be relevant for the calculation of SOE. Ultimately, the sum of all squared residuals is equal to 7,944.4. And now oops. Going back to our equation. S of E is equal to the square root of 7,944.4 Divided by 10, subtracted by 2, which gives you 8. And now this is approximately equal to 31.51. So, in this context, S E equals 35 or 1.51 tells you that the typical difference between actual price and predicted price is about $31,510. And this corresponds to option A in the multiple choice. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

Key Concepts

Coefficient of Determination (R²)

Standard Error of Estimate (s_e)

Regression Line

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