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Fit \(y=a+x\) to the data $$ \begin{array}{l|lll} \mathrm{x} & 0 & 1 & 2 \\ \hline \mathrm{y} & 1 & 3 & 4 \end{array} $$ by the method of least squares.

Short Answer

Expert verified
The line that best fits the given data by the method of least squares is \(y = 1 + 1.67\).

Step by step solution

01

Calculate Summations

First, calculate the summations needed in the following calculations, which are \(\sum x\) and \(\sum y\). Here, \(\sum x = 0 + 1 + 2 = 3\) and \(\sum y = 1 + 3 + 4 = 8\)
02

Find the Mean Values

Find the mean values of x and y with the formulas \(\overline{x} =\frac{\sum x}{n}\) and \(\overline{y} =\frac{\sum y}{n}\). In this case, \(\overline{x} =\frac{3}{3} = 1\) and \(\overline{y} = \frac{8}{3} \approx 2.67\)
03

Determine the Value of 'a'

As the given line is \(y = a + x\), this can be re-written in slope-intercept form as \(y = mx + b\), with \(m=1\) and \(b=a\). By comparing to the general line equation, determine 'a' as the y-intercept which is the mean value of y minus the mean value of x multiplied by the slope. Thus, \(a =\overline{y} - m\cdot\overline{x}\), i.e., \(a = 2.67 - 1\cdot 1 = 1.67\)

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Most popular questions from this chapter

If \(A_{1}, A_{2}, \ldots, A_{k}\) are events, prove, by induction, Boole's inequality $$ P\left(A_{1} \cup A_{2} \cup \cdots \cup A_{k}\right) \leq \sum_{1}^{k} P\left(A_{i}\right) $$ Then show that $$ P\left(A_{1}^{c} \cap A_{2}^{c} \cap \cdots \cap A_{k}^{c}\right) \geq 1-\sum_{1}^{b} P\left(A_{i}\right) $$

Show that the square of a noncentral \(T\) random variable is a noncentral \(F\) random variable.

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Let \(A\) be the real symmetric matrix of a quadratic form \(Q\) in the observations of a random sample of size \(n\) from a distribution which is \(N\left(0, \sigma^{2}\right)\). Given that \(Q\) and the mean \(\bar{X}\) of the sample are independent, what can be said of the elements of each row (column) of \(\boldsymbol{A}\) ? Hint: Are \(Q\) and \(X^{2}\) independent?

Student's scores on the mathematics portion of the ACT examination, \(x\), and on the final examination in the first-semester calculus ( 200 points possible), \(y\), are given. (a) Calculate the least squares regression line for these data. (b) Plot the points and the least squares regression line on the same graph. (c) Find point estimates for \(\alpha, \beta\), and \(\sigma^{2}\). (d) Find 95 percent confidence intervals for \(\alpha\) and \(\beta\) under the usual assumptions. $$ \begin{array}{cc|cc} \hline \mathrm{x} & \mathrm{y} & \mathrm{x} & \mathrm{y} \\ \hline 25 & 138 & 20 & 100 \\ 20 & 84 & 25 & 143 \\ 26 & 104 & 26 & 141 \\ 26 & 112 & 28 & 161 \\ 28 & 88 & 25 & 124 \\ 28 & 132 & 31 & 118 \\ 29 & 90 & 30 & 168 \\ 32 & 183 & & \\ \hline \end{array} $$

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