Insurance companies and other organizations use actuarial...

Insurance companies and other organizations use actuarial tables to estimate the remaining life spans of their customers. The table below shows the predicted additional years of life for Hispanic females of various ages in the United States, according to a 2008 National Vital Statistics Report. (www.cdc.gov/nchs/ deaths.htm) $$\begin{array}{c}{\text { Age }} & {\text { Years to Live}} \\ {10} & {73.9} \\ {20} & {64} \\ {30} & {54.2} \\ {40} & {44.5} \\ {50} & {35.1} \\ {60} & {26.1} \\ {70} & {17.8} \\ {80} & {10.6} \\ {90} & {5.3} \\ {100} & {2.6}\end{array}$$ Here are the results of a re-expression. $$\begin{array}{l}{\text { Dependent variable is: sqrt(Yrstolive) }} \\ {\text { R-squared }=99.4 \% \quad s=0.19068} \\ {\text { Variable }} & {\text { Coefficient }} \\ {\text { Intercept }} & {9.6867} \\ {\text { Age }} & {-0.0797}\end{array}$$ a) Evaluate the success of the regression. b) Predict the lifespan of an 18-year-old Hispanic woman. c) Are you satisfied that your model could predict the life expectancy of a friend of yours?

Insurance companies and other organizations use actuarial tables to estimate the remaining
life spans of their customers. The table below shows the predicted additional years of life for Hispanic females of various ages in the United States, according to a 2008 National Vital Statistics Report. (www.cdc.gov/nchs/ deaths.htm)
$$\begin{array}{c}{\text { Age }} & {\text { Years to Live}} \\ {10} & {73.9} \\ {20} & {64} \\ {30} & {54.2} \\ {40} & {44.5} \\ {50} & {35.1} \\ {60} & {26.1} \\ {70} & {17.8} \\ {80} & {10.6} \\ {90} & {5.3} \\ {100} & {2.6}\end{array}$$
Here are the results of a re-expression.
$$\begin{array}{l}{\text { Dependent variable is: sqrt(Yrstolive) }} \\ {\text { R-squared }=99.4 \% \quad s=0.19068} \\ {\text { Variable }} & {\text { Coefficient }} \\ {\text { Intercept }} & {9.6867} \\ {\text { Age }} & {-0.0797}\end{array}$$
a) Evaluate the success of the regression.
b) Predict the lifespan of an 18-year-old Hispanic woman.
c) Are you satisfied that your model could predict the life expectancy of a friend of yours?

Key Concepts

Actuarial Life Tables

Actuarial tables are statistical tools that summarize life expectancy and mortality rates for different populations and ages. They are widely used in insurance and pension planning to estimate the remaining lifetime and to assess risks associated with different demographic groups.

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. It allows prediction of the dependent variable's values given new values of the independent variable(s) and provides insights into how changes in predictors affect outcomes.

Data Transformation

Data transformation involves modifying data, such as applying a square root or logarithmic function, to meet the assumptions of regression analysis or to improve model performance. Transformations can make relationships more linear and stabilize variance, which helps in obtaining more reliable estimates.

R-squared and Standard Error

R-squared is a statistical measure that represents the proportion of the variance in the dependent variable that is explained by the independent variable(s) in the model, while the standard error measures the average distance that the observed values fall from the regression line. Together, they indicate the goodness of fit and the precision of the predictions made by the model.

Prediction in Transformed Scale

When using a regression model where the dependent variable has been transformed, predictions must be made on the transformed scale. These predictions might need to be back-transformed to the original scale for interpretation. This process requires careful handling to ensure that the predictions are accurate and meaningful in the context of the original variable.

Model Generalizability and Extrapolation

Evaluating a regression model involves not only assessing how well it fits the data but also considering its ability to generalize to new cases outside the sample data. Extrapolation involves using the model to make predictions for values of the independent variable that were not part of the original data set, and caution must be exercised as the reliability of predictions can diminish when applied to different populations or extremes.

Transcript

Please give Ace some feedback