# KV Basic Methods of Data Analysis Exam 2020/2021

###### tags: `Exam`

You are given the following set of univariate measurement data: X={3,5,6,9,2}.
* Calculate the sample mean:
* Calculate the median:
* Calculate the sample variance:

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A) Based on this answer the following:

* What is the shortest preparation time? ... days
* What longest preparation time? ... days
* What is the lower quartile? ... days
* What is the interquartile range? ... days
* What is the median time? ... days

B) How can a histogram of 16000 randomly drawn students, that respects the characteristics from the above boxplot, look like? Use a bin size of 0.5 days. Plot the density with the highest entropy, that fitts to the boxplot. This is the one that has "flattest" graph. In other words: assume that the distribution within every of the 4 quartiles is flat. "Draw" the histogram by deciding which number of students has to fit in the corresponding bins:

* Bin between 0.5 and 1: 2000 students
* Bin between 1 and 1.5: 2000 students
* Bin between 1.5 and 2: 4000 students
* Bin between 2 and 2.5: 2000 students
* Bin between 2.5 and 3: 2000 students
* Bin between 3 and 3.5: 1000 students
* Bin between 3.5 and 4: 1000 students
* Bin between 4 and 4.5: 1000 students
* Bin between 4.5 and 5: 1000 students
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* What is the dimension of the column space of A?
* What is the dimension of the row space of A?
* What is the dimension of the kernel (=null space) ker(A)?
* What is the rank of A?
* Consider an element x∈ker(A) that assumes the value x1=4 on its first coordinate. Compute the values of x2= and x3=

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* $A$
* $A^{T}A$

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A test predicts with 99.9% probability a positive test result for a HIV infected person. With 99.8% probability it yields a negative result for a person which is not infected. In total, 0.05% of all Austrian people are infected. Answer the following questions:

* How many of 1000 tested people who are infected get a negative test result (false negative)?
* How many out of 1000 non HIV-infected people wrongly get a positive result (false positive)?
* A random person takes the test and its result is positive. How large is the probability that the person is not infected? Your answer should be a number between 0 and 1 rounded up to two decimals.

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* Determine the value of the associated Maximum-Likelihood-estimator θ:

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* Compute the type I error she might make:
* Compute the test score z=
* What level p of the corresponding p-quantile does she have to look up in a table? p=
* Assume that the desired quantile computes as 2.32. What is her decision about the null-hypothesis?

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Let A,B∈Rn×n and C=AB. Tick the correct statements.

Select one or more:
- [ ] a. The p-value gives the significance level, for which the test score z lies exactly on the border between accepting and rejecting $H_0$.
- [ ] b. If the sample correlation coefficient r=−1 then all samples lie on a straight line with slope 1.
- [ ] c. If A,B are positive definite, then the same holds for $\frac{A+B}{2}$.
- [ ] d. If B is invertible, then rank(A)=rank( C ).
- [ ] e. The determinant of 3A is 3 times the determinant of A.
- [ ] f. The rank of C equals the sum of the rank of A and the rank of B.
- [ ] g. If A is a projection matrix, it can only have {−1,1} as possible eigenvalues.
- [ ] h. For two events A1 and A2 over a universe $Ω$ we always have $P(A1∪A2)=P(A1)+P(A2)$.

# Retry

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