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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

In Example \(9.1 .2\) verify that \(Q=Q_{3}+Q_{4}\) and that \(Q_{3} / \sigma^{2}\) has a chi-square distribution with \(b(a-1)\) degrees of freedom.

Let the independent random variables \(Y_{1}, \ldots, Y_{n}\) have the joint pdf. $$ L\left(\alpha, \beta, \sigma^{2}\right)=\left(\frac{1}{2 \pi \sigma^{2}}\right)^{n / 2} \exp \left\\{-\frac{1}{2 \sigma^{2}} \sum_{1}^{n}\left[y_{i}-\alpha-\beta\left(x_{i}-\bar{x}\right)\right]^{2}\right\\} $$ where the given numbers \(x_{1}, x_{2}, \ldots, x_{n}\) are not all equal. Let \(H_{0}: \beta=0(\alpha\) and \(\sigma^{2}\) unspecified). It is desired to use a likelihood ratio test to test \(H_{0}\) against all possible alternatives. Find \(\Lambda\) and see whether the test can be based on a familiar statistic. Hint: In the notation of this section show that $$ \sum_{1}^{n}\left(Y_{i}-\hat{\alpha}\right)^{2}=Q_{3}+\widehat{\beta}^{2} \sum_{1}^{n}\left(x_{i}-\bar{x}\right)^{2} $$

Using the notation of this section, assume that the means satisfy the condition that \(\mu=\mu_{1}+(b-1) d=\mu_{2}-d=\mu_{3}-d=\cdots=\mu_{b}-d .\) That is, the last \(b-1\) means are equal but differ from the first mean \(\mu_{1}\), provided that \(d \neq 0\). Let independent random samples of size \(a\) be taken from the \(b\) normal distributions with common unknown variance \(\sigma^{2}\). (a) Show that the maximum likelihood estimators of \(\mu\) and \(d\) are \(\hat{\mu}=\bar{X} . .\) and $$ \hat{d}=\frac{\sum_{j=2}^{b} \bar{X}_{. j} /(b-1)-\bar{X}_{.1}}{b} $$ (b) Using Exercise \(9.1 .3\), find \(Q_{6}\) and \(Q_{7}=c \hat{d}^{2}\) so that, when \(d=0, Q_{7} / \sigma^{2}\) is \(\chi^{2}(1)\) and $$ \sum_{i=1}^{a} \sum_{j=1}^{b}\left(X_{i j}-\bar{X}_{n}\right)^{2}=Q_{3}+Q_{6}+Q_{7} $$ (c) Argue that the three terms in the right-hand member of Part (b), once divided by \(\sigma^{2}\), are independent random variables with chi-square distributions, provided that \(d=0\). (d) The ratio \(Q_{7} /\left(Q_{3}+Q_{6}\right)\) times what constant has an \(F\) -distribution, provided that \(d=0\) ? Note that this \(F\) is really the square of the two-sample \(T\) used to test the equality of the mean of the first distribution and the common mean of the other distributions, in which the last \(b-1\) samples are combined into one.

In Exercise 9.2.1, show that the linear functions \(X_{i j}-X_{. j}\) and \(X_{. j}-X\).. are uncorrelated. Hint: Recall the definition of \(\bar{X}_{. j}\) and \(\bar{X}_{. .}\) and, without loss of generality, we can let \(E\left(X_{i j}\right)=0\) for all \(i, j\)

Show that \(\sum_{i=1}^{n}\left[Y_{i}-\alpha-\beta\left(x_{i}-\bar{x}\right)\right]^{2}=n(\hat{\alpha}-\alpha)^{2}+(\hat{\beta}-\beta)^{2} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}+\sum_{i=1}^{n}\left[Y_{i}-\hat{\alpha}-\hat{\beta}\left(x_{i}-\bar{x}\right)\right]^{2} .\)

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