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Fit y=a+x to the data x012y134 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 x and y. Here, x=0+1+2=3 and y=1+3+4=8
02

Find the Mean Values

Find the mean values of x and y with the formulas x=xn and y=yn. In this case, x=33=1 and y=832.67
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=ymx, i.e., a=2.6711=1.67

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Most popular questions from this chapter

In Example 9.1.2 verify that Q=Q3+Q4 and that Q3/σ2 has a chi-square distribution with b(a1) degrees of freedom.

Let the independent random variables Y1,,Yn have the joint pdf. Missing \left or extra \right where the given numbers x1,x2,,xn are not all equal. Let H0:β=0(α and σ2 unspecified). It is desired to use a likelihood ratio test to test H0 against all possible alternatives. Find Λ and see whether the test can be based on a familiar statistic. Hint: In the notation of this section show that 1n(Yiα^)2=Q3+β^21n(xix¯)2

Using the notation of this section, assume that the means satisfy the condition that μ=μ1+(b1)d=μ2d=μ3d==μbd. That is, the last b1 means are equal but differ from the first mean μ1, provided that d0. Let independent random samples of size a be taken from the b normal distributions with common unknown variance σ2. (a) Show that the maximum likelihood estimators of μ and d are μ^=X¯.. and d^=j=2bX¯.j/(b1)X¯.1b (b) Using Exercise 9.1.3, find Q6 and Q7=cd^2 so that, when d=0,Q7/σ2 is χ2(1) and i=1aj=1b(XijX¯n)2=Q3+Q6+Q7 (c) Argue that the three terms in the right-hand member of Part (b), once divided by σ2, are independent random variables with chi-square distributions, provided that d=0. (d) The ratio Q7/(Q3+Q6) 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 b1 samples are combined into one.

In Exercise 9.2.1, show that the linear functions XijX.j and X.jX.. are uncorrelated. Hint: Recall the definition of X¯.j and X¯.. and, without loss of generality, we can let E(Xij)=0 for all i,j

Show that i=1n[Yiαβ(xix¯)]2=n(α^α)2+(β^β)2i=1n(xix¯)2+i=1n[Yiα^β^(xix¯)]2.

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