Residual Plot Residuals (years) -100 0 100 20 40 60 Length (m) Density Plot of Residuals Boston Marathon 2010, Female Runners Density 0.000 0.005 0.010 0.015 0.020 -100 0 100 Residuals
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**Step 2:** There is evidence of strong linearity between 'Age' and 'Time' as the residual plot shows a pattern where the residuals are centered around zero, suggesting a linear relationship between the variables. **Step 3:** There is no evidence that the Show more…
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Random Residuals In a least-squares regression model, the residuals are assumed to be random. The following data represent the life expectancy of a male born in the given year (for example, a male born in 1996 is expected to live 73.1 years). $$\begin{array}{lc} \text { Year, } \boldsymbol{x} & \text { Life Expectancy, } \boldsymbol{y} \\ \hline 1996 & 73.1 \\ \hline 1997 & 73.6 \\ \hline 1998 & 73.8 \\ \hline 1999 & 73.9 \\ \hline 2000 & 74.1 \\ \hline 2001 & 74.2 \end{array}$$ $$\begin{array}{|lc|} \hline \text { Year, } \boldsymbol{x} & \text { Life Expectancy, } \boldsymbol{y} \\ \hline 2002 & 74.3 \\ \hline 2003 & 74.5 \\ \hline 2004 & 74.9 \\ \hline 2005 & 74.9 \\ \hline 2006 & 75.1 \\ \hline 2007 & 75.4 \\ \hline \end{array}$$ The least-squares regression equation treating year as the independent variable is $\hat{y}=0.1846 x-295.1910 .$ The residuals from left to right are $$\begin{array}{lrrrrr} -0.171 & 0.145 & 0.160 & 0.076 & 0.091 & 0.006 \\ \hline-0.078 & -0.063 & 0.153 & -0.032 & -0.017 & 0.099 \end{array}$$ (a) Denote residuals above zero with an A and those below zero with a B to form a sequence. (b) Test the assumption that the residuals are random at the $\alpha=0.05$ level of significance.
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Random Residuals In a least-squares regression model, the residuals are assumed to be random. The following data represent the life expectancy of a female born in the given year (for example- a female born in 1996 is expected to live 79.1 years). $$\begin{array}{lc} \text { Year, } \boldsymbol{x} & \text { Life Expectancy, } \boldsymbol{y} \\ \hline 1996 & 79.1 \\ \hline 1997 & 79.4 \\ \hline 1998 & 79.5 \\ \hline 1999 & 79.4 \\ \hline 2000 & 79.3 \\ \hline 2001 & 79.4 \end{array}$$ $$\begin{array}{|lc|} \hline \text { Year, } \boldsymbol{x} & \text { Life Expectancy, } \boldsymbol{y} \\ \hline 2002 & 79.5 \\ \hline 2003 & 79.6 \\ \hline 2004 & 79.9 \\ \hline 2005 & 79.9 \\ \hline 2006 & 80.2 \\ \hline 2007 & 80.4 \\ \hline \end{array}$$ The least-squares regression equation treating the year as the independent variable is $\hat{y}=0.0972 x-114.9181 .$ The residuals from left to right are (a) Denote residuals above 0 with an $A$ and those below 0 with a B to form a sequence. (b) Test the assumption that the residuals are random at the $\alpha=0.05$ level of significance.
Below are a scatterplot of residuals by fitted values, and a histogram and Q-Q plot of the residuals. Why do researchers look at these plots of the residuals? (check all that apply): A. To investigate whether the data has an underlying structure or pattern, which may indicate the observations are not independent. B. To judge whether the variability of the data about the regression line is nearly constant. C. To measure the variability of the slope and intercept of the model. D. To make sure the residuals follow a linear trend. E. To determine if the residuals are roughly normally distributed. F. To explain the strength of the evidence of a linear relationship between the residuals and fitted values.
Ana Carolina D.
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