# Basic Methods of Data Analysis Exam WS2019
###### tags: `Exam`


A) 5

B) 5

C) 7.5

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A)

| X | absolute frequency | relative frequency |
| --- |:------------------:|:------------------:|
| 1 | 1 | $\frac{1}{11}$ |
| 2 | 3 | $\frac{3}{11}$ |
| 3 | 5 | $\frac{5}{11}$ |
| 4 | 2 | $\frac{2}{11}$ |

B) Mode: 3 Range: 3

C)
Q1: 2

MED: 3

Q3: 3

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A) 0.8
B) 0,06

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A)
$\begin{pmatrix}2&1&0\\ 0&2&4\\ 1&3&5\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0\end{pmatrix}$
$x=z=k, y=-2k$
The are linearly dependent.

B) 2)

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A)
$\begin{pmatrix}2&4\\ \:\:1&3\end{pmatrix}\cdot \:x=\begin{pmatrix}2\\ \:\:3\end{pmatrix}$
$\mathrm{Multiply\:both\:sides\:of\:the\:equation\:by}\:\begin{pmatrix}2&4\\ 1&3\end{pmatrix}^{-1}\:\mathrm{from\:the\:left}$
$AX=B\quad \Rightarrow \quad \:X=A^{-1}B$
$x=\begin{pmatrix}2&4\\ 1&3\end{pmatrix}^{-1}\begin{pmatrix}2\\ 3\end{pmatrix}$
$x=\begin{pmatrix}-3\\ 2\end{pmatrix}$

B) L = {{1, 0}, {1/2, 1}} U={{2, 4}, {0, 1}}

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A)
* T
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B)
* F B->A
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$\mathrm{Find\:the\:eigenvalues\:for\:}\begin{pmatrix}3&-1\\ -1&3\end{pmatrix}$
$\mathrm{The\:eigenvalues\:of}\:A\:\mathrm{are\:the\:roots\:of\:the\:characteristic\:equation}\:\det \left(A-λ\:I\right)=0$
$\mathrm{The\:eigenvalues\:are:\:} λ=4,\:λ=2$

$\mathrm{Find\:the\:eigenvectors\:for\:}\begin{pmatrix}3&-1\\ -1&3\end{pmatrix}$
$\mathrm{To\:find\:the\:eigenvectors}\:η,\:\mathrm{solve\:}\left(A-λ\:I\right)η=0\mathrm{\:for\:each\:eigenvalue}\:λ$
$=\begin{pmatrix}-1\\ 1\end{pmatrix},\:\begin{pmatrix}1\\ 1\end{pmatrix}$

$\mathrm{Find\:the\:eigenvalues\:for\:}\begin{pmatrix}36&-28\\ -28&36\end{pmatrix}$
$\mathrm{The\:eigenvalues\:of}\:A\:\mathrm{are\:the\:roots\:of\:the\:characteristic\:equation}\:\det \left(A-λ\:I\right)=0$
$\mathrm{The\:eigenvalues\:are:} \: λ=64,\:λ=8$

$\mathrm{Find\:the\:eigenvectors\:for\:}\begin{pmatrix}36&-28\\ -28&36\end{pmatrix}$
$\mathrm{To\:find\:the\:eigenvectors}\:η,\:\mathrm{solve\:}\left(A-λ\:I\right)η=0\mathrm{\:for\:each\:eigenvalue}\:λ$
$=\begin{pmatrix}-1\\ 1\end{pmatrix},\:\begin{pmatrix}1\\ 1\end{pmatrix}$

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A) 0,4

B) 0,6

C) $\frac{4}{9}$ are dependent (?)

D) $\frac{6}{9}$ are dependent (?)

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A) 0,3456

B) E(head)=np=0.6*5=3

C) E(tails)=np=0.4*5=2

D) 1,2 (?)

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A)

B) $Z = \frac{X-µ_{0}}{S} \cdot \sqrt{n} = -2$

C) $|Z| > 0,995$ -> reject Ho

D)

E)

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* \
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* F {0,1}
* T
* F $P(A) + P(B) - P(A \cup B)$
* T
*
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