Table 12.20 shows the life expectancy for an individual born...

Table 12.20 shows the life expectancy for an individual born in the United States in certain years. $$\begin{array}{|c|c|}\hline \text { Year of Birth } & {\text { Life Expectancy }} \\ \hline 1930 & {59.7} \\ \hline 1940 & {62.9} \\ \hline 1950 & {70.2} \\ \hline 1950 & {71.5} \\ \hline 1987 & {75} \\ \hline 1987 & {75.7} \\ \hline 2010 & {78.7} \\ \hline\end{array}$$ a. Decide which variable should be the independent variable and which should be the dependent variable. b. Draw a scatter plot of the ordered pairs. c. Calculate the least squares line. Put the equation in the form of: $\hat{y}=a+b x$ d. Find the correlation coefficient. Is it significant? e. Find the estimated life expectancy for an indidual born in 1950 and for one born in 1982 f. Why aren't the answers to part e the same as the values in Table 12.20 that correspond to those years? g. Use the two points in part e to plot the least squares line on your graph from part b. h. Based on the data, is there a linear relationship between the year of birth and life expectancy? i. Are there any outliers in the data? j. Using the least squares line, find the estimated life expectancy for an individual born in 1850. Does the least squares line give an accurate estimate for that year? Explain why or why not. k. What is the slope of the least-squares (best-fit) line? Interpret the slope.

Table 12.20 shows the life expectancy for an individual born in the United States in certain years.
$$\begin{array}{|c|c|}\hline \text { Year of Birth } & {\text { Life Expectancy }} \\ \hline 1930 & {59.7} \\ \hline 1940 & {62.9} \\ \hline 1950 & {70.2} \\ \hline 1950 & {71.5} \\ \hline 1987 & {75} \\ \hline 1987 & {75.7} \\ \hline 2010 & {78.7} \\ \hline\end{array}$$
a. Decide which variable should be the independent variable and which should be the dependent variable.
b. Draw a scatter plot of the ordered pairs.
c. Calculate the least squares line. Put the equation in the form of: $\hat{y}=a+b x$
d. Find the correlation coefficient. Is it significant?
e. Find the estimated life expectancy for an indidual born in 1950 and for one born in 1982
f. Why aren't the answers to part e the same as the values in Table 12.20 that correspond to those years?
g. Use the two points in part e to plot the least squares line on your graph from part b.
h. Based on the data, is there a linear relationship between the year of birth and life expectancy?
i. Are there any outliers in the data?
j. Using the least squares line, find the estimated life expectancy for an individual born in 1850. Does the least squares line give an accurate estimate for that year? Explain why or why not.
k. What is the slope of the least-squares (best-fit) line? Interpret the slope.

Key Concepts

Correlation Coefficient

The correlation coefficient is a statistical measure that quantifies the degree to which two variables are linearly related. It ranges from -1 to 1, with values closer to either extreme indicating a stronger linear relationship. It is also used to assess the significance of the relationship between the variables.

Independent vs. Dependent Variables

In data analysis, the independent variable (predictor) is the one that is assumed to influence or predict the other variable, while the dependent variable (response) is the one that is presumed to depend on the independent variable. Deciding which variable plays which role is crucial in setting up regression models and analyzing their relationship.

Scatter Plot

A scatter plot is a graphical representation of the relationship between two quantitative variables. It involves plotting ordered pairs (x, y) on a coordinate system, which helps to visually assess the nature, strength, and direction of any linear or non-linear relationship between the variables.

Least Squares Regression Line

The least squares regression line is a method used to fit a straight line to a set of data points such that the sum of the squares of the vertical distances (residuals) of the points from the line is minimized. This line provides a model for predicting the dependent variable from the independent variable.

Prediction Using Regression Models

Once a regression model is developed, it can be used to estimate or predict the value of the dependent variable for given values of the independent variable. However, predictions may differ from actual observed values due to natural variability, measurement error, or model approximation, and caution should be taken particularly when predicting values outside the range of the observed data (extrapolation).

Outliers

Outliers are data points that differ significantly from other observations. They can greatly affect the results of a regression analysis, potentially skewing the best-fit line and impacting the correlation coefficient. Identifying and understanding outliers is an important step in data analysis.

Extrapolation and its Limitations

Extrapolation is the process of using a regression model to predict values outside the range of the observed data. Because the relationship modeled may not hold beyond the observed range, extrapolated predictions are often less reliable and can be inaccurate if the underlying assumptions of the regression are violated.

Slope of the Least Squares Line

The slope of the least squares line represents the rate of change of the dependent variable with respect to the independent variable. It tells us how much the dependent variable is expected to increase (or decrease) for each one-unit increase in the independent variable, providing a clear interpretation of the relationship between the two variables.

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