91Ó°ÊÓ

The following data on degree of exposure to \({ }^{242} \mathrm{Cm}\) alpha particles \((x)\) and the percentage of exposed cells without aberrations \((y)\) appeared in the paper "Chromosome Aberrations Induced in Human Lymphocytes by D-T Neutrons" (Radiation Research \([1984]: 561-573)\) : $$ \begin{array}{rrrrr} x & 0.106 & 0.193 & 0.511 & 0.527 \\ y & 98 & 95 & 87 & 85 \\ x & 1.08 & 1.62 & 1.73 & 2.36 \\ y & 75 & 72 & 64 & 55 \\ x & 2.72 & 3.12 & 3.88 & 4.18 \\ y & 44 & 41 & 37 & 40 \end{array} $$ Summary quantities are $$ \begin{gathered} n=12 \quad \sum x=22.027 \quad \sum y=793 \\ \sum x^{2}=62.600235 \quad \sum x y=1114.5 \quad \sum y^{2}=57,939 \end{gathered} $$ a. Obtain the equation of the least-squares line. b. Construct a residual plot, and comment on any interesting features.

Step by step solution

01

Calculate the slope of the regression line

We will calculate the slope (b) for the least squares line using the formula: \[b=\frac{n\(\sum xy\)-\(\sum x\)\(\sum y\)}{n\(\sum x^{2}\)-\(\sum x\)^{2}}\]. We can use the summary quantities given in the problem to solve for b. Substituting the values, we get \[b=\frac{12(1114.5)-22.027(793)}{12(62.600235)-(22.027^{2})}\]. We can calculate this numerical value.

02

Calculate the y-intercept of the regression line

After finding the slope, we will calculate the y-intercept (a) for the least squares line using the formula: \[a=\frac{\sum y-b\(\sum x\)}{n}\]. Again, substituting the known values, we get \[a= \frac{793 - b(22.027)}{12}\]. Calculate this value as well.

03

Frame the equation of the least squares line

Next, we can construct the equation of the least squares line. The general form of the equation is \(y= a + bx\). So our equation of the least squares line will be based on the 'a' and 'b' values found earlier. Substitute 'a' and 'b' with the calculated numerical values.

04

Construct the residual plot

To construct a residual plot, take the observed values of y from the data and the predicted values of y (calculated from the equation of the line) . Subtract the predicted y-value from each observed y-value to get the residuals. Then, create a scatter plot of the residuals against the observed x-values. Note any patterns or unusual observations.

05

Comment on the residual plot

The ideal residual plot should display a 'random scatter', meaning the points should not follow any specific pattern but should be randomly distributed around the horizontal line y=0. If there's any pattern like curvature, it means the relationship between x and y is not linear. Similarly, if the scatter is much wider at one end of the graph than at the other end, it means residuals' variance is not constant.

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Key Concepts

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

Least Squares Method

The Least Squares Method is a popular technique for finding the line of best fit in a set of data points. It minimizes the sum of the squares of the differences between the observed values and the values predicted by the line, also known as residuals. This method provides us with the most accurate linear approximation of the data. By using the given formula for the slope and y-intercept, we find the regression line equation, which helps in predicting values and understanding the data trend.

• First, calculate the slope \(b\) using the formula: \[ b=\frac{n\(\sum xy\) - \(\sum x\)\(\sum y\)}{n\(\sum x^{2}\) - \(\sum x\)^{2}} \].
• Assume substitute known values to compute \(b\).
• Determine the y-intercept \(a\) using: \[ a= \frac{\sum y - b\(\sum x\)}{n} \]. After obtaining the slope \(b\) and intercept \(a\), plug these into the linear equation \(y = a + bx\).

This final equation represents the line that best describes the relationship between the two variables in your dataset.

Residual Plot

Creating a residual plot is crucial to evaluate the validity of the least squares regression line. A residual plot displays the residuals on the y-axis and the independent variable \(x\) on the x-axis. Residuals are calculated as the difference between the observed and predicted values of \(y\). The purpose of this plot is to check for randomness in the distribution of these residuals.

• A random scatter of points along the horizontal line \(y=0\) indicates a well-fitting linear model.
• Patterns, such as curves, suggest a non-linear relationship between the variables.
• Trends in spread (non-constant variance) indicate issues like heteroscedasticity, which may affect the predictive accuracy of the model.

A good residual plot ensures the linear equation is appropriate for the data, confirming that the relationship captured is indeed linear.

Linear Equation

A linear equation describes a straight line graph and has the general form \(y = a + bx\). In regression analysis, this equation predicts the dependent variable \(y\) based on the independent variable \(x\). For this exercise:

• \(a\) is the y-intercept, showing where the line crosses the y-axis when \(x=0\).
• \(b\) is the slope, indicating the rate at which \(y\) changes as \(x\) increases or decreases.

Creating this equation is the goal of regression analysis since it allows us to make predictions and understand the nature of the relationship between two continuous variables. Once the slope and intercept are calculated, using them will let you comprehend and predict future data trends.

Data Interpretation

Interpreting data in the context of a linear regression involves understanding the numerical output of the least squares line and applying it to real-world scenarios. Here are the key steps:

• Examine the slope \(b\). If it's positive, it indicates that \(y\) increases as \(x\) increases. Conversely, if it's negative, \(y\) decreases as \(x\) increases.
• The y-intercept \(a\) determines the starting point of the line on the y-axis.
• The quality of the linear relationship is often expressed as the strength and direction of the correlations observed in the data.

Correct interpretation provides insight into the dynamics of the studied variables, helping in making informed decisions based on the results of your analysis. Always validate analysis using visual aids like graphs to avoid misinterpretations related to hidden patterns or non-constant variance.