Quiz Chapter 06.03: Linear Regression – Holistic Numerical Methods

1. Given $\left(x_{1},y_{1} \right), \, \left( x_{2}, y_{2} \right), \, ... \, , \, \left( x_{n}, y_{n} \right)$ best fitting data to $y = f (x)$ by least squares requires minimization of

\, \displaystyle\sum_{i = 1}^{n} \left[ y_{i} - f \left( x_{i} \right) \right]

\, \displaystyle\sum_{i = 1}^{n} \left| y_{i} - f \left( x_{i} \right) \right|

\, \displaystyle\sum_{i = 1}^{n} \left[ y_{i} - f \left( x_{i} \right) \right]^{2}

\, \displaystyle\sum_{i = 1}^{n} \left[ y_{i} - \bar{y} \right]^{2}, \, \bar{y} = \dfrac{ \displaystyle\sum_{i = 1}^{n} y_{i}}{n}

2. The following data

$x$	$1$	$20$	$30$	$40$
$y$	$1$	$400$	$800$	$1300$

is regressed with least square regression to $y = a_{0} + a_{1}x$ . The value of $a_{1}$ is most nearly

27.480

28.956

32.625

40.000

3. The following data

$x$	$1$	$20$	$30$	$40$
$y$	$1$	$400$	$800$	$1300$

is regressed with least square regression to $y = a_{1}x$ . The value of $a_{1}$ is most nearly

27.480

28.956

32.625

40.000

4. An instructor gives the same $y$ vs $x$ data as given below to four students and asks them to regress the data with least squares regression to $y = a_{0} + a_{1}x$ .

$x$	$1$	$10$	$20$	$30$	$40$
$y$	$1$	$100$	$400$	$600$	$1200$

Each student comes up with four different answers for the straight-line regression model. Only one is correct. The correct model is

y = 60x - 1200

y = 30x - 200

y = -139.43 + 29.684x

y = 1 + 22.732x

5. A torsion spring of a mousetrap is twisted through an angle of $180^{\circ}$ . The torque vs. angle data is given below

Torsion, $T$ , N-m	$0.110$	$0.189$	$0.230$	$0.250$
Angle, $\theta$ , rad	$0.10$	$0.50$	$1.1$	$1.5$

The amount of strain energy stored in the mousetrap spring in Joules is

0.29872

0.41740

0.84208

1561.8

6. A scientist finds that regressing the $y$ vs $x$ data given below to $y = a_{0} + a_{1}x$ results in the coefficient of determination for the straight-line model, $r^{2}$ , being zero

$x$	$1$	$3$	$11$	$17$
$y$	$2$	$6$	$22$	?

The missing value for $y$ at $x = 17$ most nearly is

-2.4444

2.0000

6.8889

34.000