Suppose 5 out of 25 data points in a weighted

Find the least-squares line \(y=\beta_{0}+\beta_{1} x\) that best fits the data (-2,0),(-1,0),(0,2),(1,4), and (2,4), assuming that the first and last data points are less reliable. Weight them half as much as the three interior points.

Problem 2E Suppose 5 out of 25 data points in a weighted least-squares problem have a y-measurement that is less reliable than the others, and they are to be weighted half as much as the other 20 points. One method is to weight the 20 points by a factor of 1 and the other 5 by a factor of . A second method is to weight the 20 points by a factor of 2 and the other 5 by a factor of 1. Do the two methods produce different results? Explain.

Problem 16E [M] Let f4 and f5 be the fourth-order and fifth-order Fourier approximations in to the square wave function in Exercise 10. Produce separate graphs of f4 and f5 on the interval , and produce a graph of f5 on . Reference Exercise 10: In Exercises 5–14, the space is with the inner product (6). Find the third-order Fourier approximation to the square wave function,

Problem 3E Fit a cubic trend function to the data in Example 2. The orthogonal cubic polynomial is Reference Example 2: The simplest and most common use of trend analysis occurs when the points can be adjusted so that they are evenly spaced and sum to zero. Fit a quadratic trend function to the data

Problem 5E In Exercises 5–14, the space is with the inner product (6). Show that sin mt and sin nt are orthogonal when m ? n.

Problem 4E To make a trend analysis of six evenly spaced data points, one can use orthogonal polynomials with respect to evaluation at the points and 5. a. Show that the first three orthogonal polynomials are (The polynomial p2 has been scaled so that its values at the evaluation points are small integers.) b. Fit a quadratic trend function to the data

In Exercises 5–14, the space is \(C[0,2 \pi]\) with the inner product (6). Show that sin mt and cos nt are orthogonal for all positive integers m and n.

Problem 7E In Exercises 5–14, the space is with the inner product (6). Show that

Problem 9E In Exercises 5–14, the space is with the inner product (6). Find the third-order Fourier approximation

In Exercises 5–14, the space is \(C[0,2 \pi]\) with the inner product (6). Find the third-order Fourier approximation to \(\sin ^{2} t\), without performing any integration calculations.

Problem 10E In Exercises 5–14, the space is with the inner product (6). Find the third-order Fourier approximation to the square wave function,

Problem 12E In Exercises 5–14, the space is with the inner product (6). Find the third-order Fourier approximation to cos3 t , without performing any integration calculations.

In Exercises 5–14, the space is \(C[0,2 \pi]\) with the inner product (6). Suppose the first few Fourier coefficients of some function f in \(C[0,2 \pi]\) are \(a_{0}, a_{1}, a_{2}, \text { and } b_{1}, b_{2}, b_{3}\). Which of the following trigonometric polynomials is closer to f ? Defend your answer. \(g(t)=\frac{a_{0}}{2}+a_{1} \cos t+a_{2} \cos 2 t+b_{1} \sin t\) \(h(t)=\frac{a_{0}}{2}+a_{1} \cos t+a_{2} \cos 2 t+b_{1} \sin t+b_{2} \sin 2 t\)

[M] Refer to the data in Exercise 13 in Section 6.6, concerning the takeoff performance of an airplane. Suppose the possible measurement errors become greater as the speed of the airplane increases, and let W be the diagonal weighting matrix whose diagonal entries are 1, 1, 1, .9, .9, .8, .7, .6, .5, .4, .3, .2, and .1. Find the cubic curve that fits the data with minimum weighted least-squares error, and use it to estimate the velocity of the plane when t = 4:5 seconds.

In Exercises 5–14, the space is \(C[0,2 \pi]\) with the inner product (6). Find the third-order Fourier approximation to f(t) = t - 1.

Problem 13E In Exercises 5–14, the space is with the inner product (6). Explain why a Fourier coefficient of the sum of two functions is the sum of the corresponding Fourier coefficients of the two functions.