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The accompanying data on \(x=\) U.S. population (millions) and \(y=\) crime index (millions) appeared in the article "The Normal Distribution of Crime" (Journal of Police Science and Administration \([1975]: 312-318)\). The author comments that "The simple linear regression analysis remains one of the most useful tools for crime prediction." When observations are made sequentially in time, the residuals or standardized residuals should be plotted in time order (that is, first the one for time \(t=1\) ( 1963 here), then the one for time \(t=2\), and so on ). Notice that here \(x\) increases with time, so an equivalent plot is of residuals or standardized residuals versus \(x\). Using \(\hat{y}=47.26+.260 x\), calculate the residuals and plot the \((x\), residual) pairs. Does the plot exhibit a pattern that casts doubt on the appropriateness of the simple linear regression model? Explain. \(\begin{array}{lrrrrrr}\text { Year } & 1963 & 1964 & 1965 & 1966 & 1967 & 1968 \\ x & 188.5 & 191.3 & 193.8 & 195.9 & 197.9 & 199.9 \\ y & 2.26 & 2.60 & 2.78 & 3.24 & 3.80 & 4.47 \\\ \text { Year } & 1969 & 1970 & 1971 & 1972 & 1973 & \\ x & 201.9 & 203.2 & 206.3 & 208.2 & 209.9 & \\ y & 4.99 & 5.57 & 6.00 & 5.89 & 8.64 & \end{array}\)

Step by step solution

01

Calculate the Residuals

The first step involves calculating the residuals for each year which is the difference between actual crime index \(y\) and predicted crime index \(\hat{y}\). The predicted crime index (\(\hat{y}\)) is calculated using the given equation \(\hat{y}=47.26+.260*x\). The residual is calculated as Residual = \(y - \hat{y}\).

02

Sequential Residual Plot

Plot the residuals against the U.S. population for each year. This will help to visualize if there is any specific pattern in residuals change over time.

03

Interpret the Plot

After obtaining the plot, observe for any discernible pattern. If the residuals exhibit a clear pattern, it may imply that a linear model is unsuitable and the assumption of the relationship between y and x being linear is not correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Residual Analysis

Residual Analysis is a crucial part of evaluating how well a regression model fits the data. In simple linear regression, it involves calculating the residuals, which are the differences between the actual observed values and the values predicted by the model. This calculation helps identify how 'off' the predictions are for each data point. When residuals are plotted against the independent variable or time, you can identify patterns that may indicate issues with the model. Residuals should not exhibit any pattern if the model is appropriate. Random scatter indicates the model is adequately capturing the relationship between the variables. However, if the residuals display a trend or systematic pattern, it suggests that the model may be missing key aspects of the data, such as curvature or variance changes over time. Analyzing residuals helps in refining models and ensuring model assumptions like linearity and homoscedasticity hold.

Crime Prediction

Using Simple Linear Regression for Crime Prediction involves modeling the relationship between variables such as population size and crime rate. The goal is to predict future crime levels based on known factors. The equation form used here is \[ \hat{y} = 47.26 + 0.260x \]where \(x\) denotes the population in millions, and \(\hat{y}\) represents the predicted crime index.This formula helps in estimating how changes in population might affect crime rates. The accuracy of these predictions depends on how well the linear model describes the relationship. Making accurate predictions is crucial for planning resources and strategies to combat crime effectively. However, if the predictions consistently deviate due to unaccounted factors, the model may need refinement or augmentation with other variables.

Time Series Analysis

Time Series Analysis explores how data points are ordered in time and how this order impacts analysis. In the context of crime prediction using regression models, time series analysis can determine if the timing of observations affects residuals in a linear model. Observations should be assessed to see if patterns develop over time. If residuals from sequential years start showing a consistent pattern, such as increasing or decreasing consistently, it might suggest additional factors that vary over time are influencing crime rates. This could mean that a simple linear approach is limited. These insights could lead to considering dynamic models that incorporate time-dependent factors such as economic changes or seasonal variation. Proper time series analysis can highlight the need to adjust the model for improved prediction accuracy.

Model Appropriateness

Assessing Model Appropriateness is about determining if the chosen linear regression model is suitable for the data. This involves reviewing plots of residuals and checking for any signs of trends or patterns. For a model to be termed appropriate, the residuals should look randomly distributed with no detectable structure. It's essential to check for key assumptions:

• Linearity: The relationship between the independent and dependent variables should be linear.
• Independence: Observations should be independently sampled.
• Homoscedasticity: The variance of the residuals should be constant across all levels of the independent variable.
• Normality: Residuals should be normally distributed.

If these assumptions are violated, alternative models or transformation of variables may be required to better fit the data. By ensuring model appropriateness, predictions and insights derived from the regression become more robust and reliable.