Math  /  Data & Statistics

QuestionA property analyst wishes to investigate the relationship between the town coundi's valuation of residential property in Johannesburg and the market value (current selling price in R 100.000) of properties. A random sample of 13 recent property transactions was examined. \begin{tabular}{|c|c|} \hline Council valuation & Markel value \\ \hline 1659 & 1850 \\ \hline 1920 & 2000 \\ \hline 1900 & 1959 \\ \hline 1200 & 1400 \\ \hline 1000 & 1200 \\ \hline 850 & 1000 \\ \hline 780 & 920 \\ \hline 870 & 990 \\ \hline 915 & 1100 \\ \hline 820 & 978 \\ \hline 1300 & 1700 \\ \hline 1500 & 1900 \\ \hline 1700 & 1850 \\ \hline \end{tabular}
4. 1 Define the independent and the dependent variabies 4.2 Draw a scatter plot to portrait the relationstip between independent variables and the Gependent variabies. interpret the scatter piot 43 Find the equation of the regression line describing the best fit of the relationship 44 Calculate the coefficient of correlation
4. 5 find the cocflicient of determination A.5 Estimate the market value of properties in fohannestburg that have a counctif variation of R 1200 (in R100\mathbf{R} 100 000) 4 interpret the meaning of the cosflicient of regression coetlicent (8,4)(8,4)

Studdy Solution

STEP 1

} \begin{enumerate} \item We have two sets of data: Council valuation (independent variable, xx) and Market value (dependent variable, yy). \item We need to determine the relationship between xx and yy using statistical methods. \item We will use linear regression to find the best-fit line. \item The scatter plot will help visualize the relationship. \item The correlation coefficient measures the strength and direction of the linear relationship. \item The coefficient of determination indicates the proportion of variance in the dependent variable that is predictable from the independent variable. \end{enumerate}
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STEP 2

1. } \begin{enumerate} \item Define the independent and dependent variables. \item Create a scatter plot to visualize the relationship. \item Calculate the equation of the regression line. \item Calculate the coefficient of correlation. \item Calculate the coefficient of determination. \item Estimate the market value for a given council valuation. \item Interpret the meaning of the regression coefficient. \end{enumerate}
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STEP 3

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Define the independent variable (xx) and the dependent variable (yy).
\begin{itemize} \item Independent variable (xx): Council valuation. \item Dependent variable (yy): Market value. \end{itemize} \textbf{}
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STEP 4

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Create a scatter plot to visualize the relationship between Council valuation and Market value.
\begin{itemize} \item Plot each pair of (x,y)(x, y) values on a graph. \end{itemize}
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STEP 5

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Interpret the scatter plot.
\begin{itemize} \item Look for a pattern (positive, negative, or no correlation). \end{itemize}
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STEP 6

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Calculate the equation of the regression line.
\begin{itemize} \item Use the formula for the slope bb: \begin{equation} b = \frac{n \sum x y - \sum x \sum y}{n \sum x^2 - (\sum x)^2} \end{equation} \item Use the formula for the intercept aa: \begin{equation} a = \frac{\sum y - b \sum x}{n} \end{equation} \end{itemize}
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STEP 7

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Substitute the given values into the formulas for aa and bb and solve.
\begin{itemize} \item Calculate x\sum x, y\sum y, x2\sum x^2, y2\sum y^2, and xy\sum xy. \item Substitute these values into the formulas for aa and bb. \end{itemize}
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STEP 8

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Write the equation of the regression line in the form y=a+bxy = a + bx.
\begin{equation} y = a + bx \end{equation}
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STEP 9

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Calculate the coefficient of correlation rr.
\begin{equation} r = \frac{n \sum xy - \sum x \sum y}{\sqrt{(n \sum x^2 - (\sum x)^2)(n \sum y^2 - (\sum y)^2)}} \end{equation}
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STEP 10

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Substitute the calculated values into the formula for rr and solve.
\begin{itemize} \item Calculate the numerator and denominator. \item Solve for rr. \end{itemize}
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STEP 11

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Calculate the coefficient of determination r2r^2.
\begin{equation} r^2 = (\text{correlation coefficient})^2 \end{equation}
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STEP 12

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Estimate the market value of properties with a council valuation of R 1200.
\begin{equation} y = a + b(1200) \end{equation}
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STEP 13

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Substitute the values of aa and bb into the equation to find the estimated market value.
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STEP 14

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Interpret the meaning of the regression coefficient (slope bb).
\begin{itemize} \item The slope bb represents the change in market value for each unit change in council valuation. \end{itemize}
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\textbf{Solution:} \begin{itemize} \item The regression line is in the form y=a+bxy = a + bx. \item The coefficient of correlation indicates the strength and direction of the linear relationship. \item The coefficient of determination shows how much of the variation in the dependent variable is explained by the independent variable. \item The estimated market value for a council valuation of R 1200 can be calculated using the regression equation. \item The slope bb shows the effect of council valuation on market value. \end{itemize}

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