91Ó°ÊÓ

The following data on the relationship between degree of exposure to \({ }^{242} \mathrm{Cm}\) alpha radiation particles \((x)\) and the percentage of exposed cells without aberrations \((y)\) appeared in the paper "Chromosome Aberrations Induced in Human Lymphocytes by DT 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 $$ \begin{gathered} n=12 \quad \sum x=22.207 \quad \sum y=793 \\ \sum x^{2}=62.600235 \quad \sum x y=1114.5 \quad \sum y^{2}=57,939 \end{gathered} $$ a. Obtain the equation of the least-squares line. b. Calculate SSResid and SSTo. c. What percentage of observed variation in \(y\) can be explained by the approximate linear relationship between the two variables? d. Calculate and interpret the value of \(s_{e}\). e. Using just the results of Parts (a) and (c), what is the value of Pearson's sample correlation coefficient?

Step by step solution

01

Calculate the least-squares line

The equation for the least-squared line is given by \( y = a + b*x \) where, \( a = (\sum y * \sum x^2 - \sum x * \sum xy) / (n*\sum x^2 - (\sum x)^2) \) and \( b = (n*\sum xy - \sum x * \sum y) / (n*\sum x^2 - (\sum x)^2) \). Substitute the provided summation values into these equations to obtain values for a and b.

02

Calculate SSResid and SSTo

SSResid (sum of squares of residuals) is given by \( \sum y^2 - b * \sum xy - a * \sum y \). SSTo (total sum of squares) is given by \( \sum y^2 - (\sum y)^2 / n \). Substitute the provided summation values into these equations to obtain values for SSResid and SSTo.

03

Calculate the percentage of observed variation

This is calculated using the formula \( R^2 = 1 - (SSResid/SSTo) \). Here, \( R^2 \) is the coefficient of determination and it indicates the percentage of the variance in the dependent variable that the independent variables explain collectively.

04

Calculate the value of \( s_{e} \)

Once SSResid is calculated, use \( s_{e} = \sqrt{SSResid / (n - 2)} \). This standard error of estimate provides the standard deviation of the residuals or prediction errors.

05

Calculate Pearson's sample correlation coefficient

Given SQRT(SSTo) as denominator and SQRT(SSTo - SSResid) as numerator, use the following formula to calculate Pearson's sample correlation coefficient, \( r = \sqrt{SSTo - SSResid} / \sqrt{SSTo} \).

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Key Concepts

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

Least-Squares Line

The least-squares line, also known as the line of best fit, is crucial in linear regression analysis. It provides a straight line that best represents the data in a scatter plot by minimizing the sum of the squared differences—or residuals—between observed and predicted values. This helps students model the relationship between an independent variable, denoted as \(x\), and a dependent variable, represented as \(y\).
By applying the formulas mentioned above for \(a\) and \(b\), students can find the specific equation of the line, \(y = a + bx\). Here, \(a\) is the y-intercept, indicating where the line crosses the y-axis, while \(b\) is the slope, showing the line's steepness and direction.
Understanding this line is fundamental because it allows predictions about the dependent variable when given values for the independent one. Using the least-squares method ensures that the calculated line minimizes errors and represents the data as accurately as possible.

Correlation Coefficient

The correlation coefficient, represented as \(r\), captures the strength and direction of a linear relationship between two variables. Its value ranges from -1 to 1.
- A value near 1 suggests a strong positive linear relationship, where \(y\) tends to increase as \(x\) increases.
- A value near -1 indicates a strong negative linear relationship, where \(y\) tends to decrease as \(x\) increases.
- A value around 0 means there is little to no linear relationship between the variables.
In the context of linear regression, \(r\) not only provides information about the linear association but also serves as a foundation for calculating \(R^2\)—the coefficient of determination. A higher magnitude of \(r\) indicates a stronger linear relationship and greater predictive accuracy.
When students use the given formulas after finding \(SSResid\) and \(SSTo\), they can determine \(r\), gaining insight into how closely the model represents the observed data.

Variation Explained by Linear Relationship

The percentage of variation explained by the linear relationship is quantified as \(R^2\), or the coefficient of determination. This statistical measure indicates the proportion of the variance in the dependent variable that is predictable from the independent variable.
Specifically, in a linear regression context, \(R^2\) is calculated as \(1-(SSResid/SSTo)\). Here, \(SSResid\) (sum of squares of residuals) measures the error variance, while \(SSTo\) (total sum of squares) measures the total variance in the dependent variable data.
A higher \(R^2\) value, closer to 1, suggests that the linear relationship explains a significant portion of the variance, thus yielding a reliable model. Conversely, an \(R^2\) value closer to 0 implies that the model does not explain much variance, indicating that other factors might influence the dependent variable significantly.
Understanding \(R^2\) can help students discern how well their chosen model fits the data and guides in making better predictions.

Standard Error of Estimate

The standard error of the estimate, denoted as \(s_e\), helps quantify the accuracy of predictions made by a regression line. It represents the average distance that the observed values fall from the regression line.
Mathematically, \(s_e\) is calculated using the formula \( s_e = \sqrt{SSResid / (n - 2)} \). In this, \(n\) refers to the number of observations, and \(SSResid\) is the sum of squares of residuals.
- A smaller \(s_e\) value indicates that the data points are closer to the line of best fit, suggesting more reliable and precise predictions.
- Conversely, a larger \(s_e\) implies greater spread of data points and less precise predictions.
By understanding \(s_e\), students can evaluate the goodness-of-fit of their linear model better, assessing how much prediction error exists in their estimates.