Minimals

Question 0

Define the combinations of n different elements taken k at a time in words.

Answer 0

All the possible selections (subsets) of size k of n different elements.

Question 1

What are the permutations of n different elements taken n at a time and what is their number?

Answer 1

Permutations of n different elements taken n at a time are all the possible linear arrangements(orders) of all the elements. The number is:
n!=n(n-1)(n-2)…2x1

Question 2

What is the meaning of (n k) binomial coefficient? Define it with a formula and with reference to combinatorics (counting techniques)

Answer 2

(n k)=n! / k!(n-k)!

The (n k) binomial coefficient gives the number ways of k elements can be chosen from n different elements without regard to order.

Question 3

Define nominal scale and give an example for it.

Answer 3

Nominal scale is a list of mutually exclusive categories to which observations can be classified. E.g. The sex of a patient can be male or female.

Question 4

Define ordinal scale and give an example for it.

Answer 4

Ordinal scale is a list of categories in which categories can be ranked according to their names or numbers assigned to them. E.g. The efficiency of a drug treatment can be bad, average, good.

Question 5

Define interval scale and give an example for it.

Answer 5

The interval scale is a quantitate scale type in which the numbers assigned to observations measured on an interval scale also express quantitative relationships. However, their ratios are not meaningful due to the lack of an objective zero point on the scale. E.g. Temperature measured on the Celsius scale.

Question 6

Define ratio scale and give an example for it.

Answer 6

Ratio scale is a quantitative scale type in which the numbers assigned to observations have real quantitative meaning. Both differences between the ratios of observations measured in a ratio scale express quantitate relationships due to the presence of an objective zero point on the scale. E.g. Measurement of height or blood glucose level.

Question 7

How are the relative frequency and probability of an event related to each other?

Answer 7

The probability of an event is the number around which the relative frequency (k/n) oscillates (n-the total number of experiments; k-the number of experiments in which the event occurred). If the number of experiments is very large, the variation of relative frequency becomes negligible. This number is called he probability of the event.

Question 8

How is classical probability denied?

Answer 8

If there are N mutually exclusive and equally like outcomes of an event, and k of these poses a trait, E, the probability of E is equal to k/N.

Question 9

What kind of values can (mathematical) probability assume? What is the probability of a certain and an impossible event?

Answer 9

Probability is a number between 0 and 1, more rigorously probability can assume any value in the closed interval of [0,1]. (A closed interval includes it’s endpoint, in the above case numbers 0 and 1)
Probability of a certain event=1
Probability of an impossible event=0

Question 10

Describe the relationship between the probabilities of event A and its complement event B

Answer 10

P(A)+P(B)=1

Question 11

Define the sum of events A and B!

Answer 11

The sum of A and B is the event which occurs when either A or B or both of them occur.

Question 12

What is the probability of the sum of events A and B?

Answer 12

P(A+B)=P(A)+P(B)-P(AB)

Where A+B is the sum of events A and B, AB is the product of events A and B.

Question 13

Define the products of event A and B!

Answer 13

The product of A and B is the event which occurs when both A and B occur.

Question 14

Define the compliment event of event A!

Answer 14

The compliment of A is the event which occurs when A doesn’t occur and the sum of the probabilities of A and its compliment event is 1.

Question 15

When are events A and B exclusive?

Answer 15

If AB=0.

Question 16

When are events A and B independent of each other?

Answer 16

A and B are independent if event B has no effect on the probability of A and vice versa.
I.E. P(AB)=P(A)xP(B) or
P(A|B)=P(A) or
P(B|A)=P(B)

Question 17

Using the terms of set theory define
A. The product of events A and B
B. The sun of events A and B

Answer 17

A. AB- the intersection of events A and B. A+B- the Union of events A and B (AUB)

Question 18

What is the meaning of P(A|B)?

Answer 18

P(A|B) is the conditional probability of A given B, I. E. The probability of occurrence of A if only those cases are considered when B occurs.

Question 19

What is the definition of a random variable?

Answer 19

If the values assumed by a variable are determined by chance factors, I. E. they can’t be exactly predicted in advance, the variable is called a random variable.

Question 20

When can a random variable be defined as continuous?

Answer 20

A random variable is continuous if it can assume any value within a specified interval of values.

Question 21

What is the definition of the cumulative frequency distribution function or cumulative relative frequency of a sample?

Answer 21

Cumulative frequency distribution function of a sample at x gives the fraction of elements in the sample which are smaller than or equal to x.

Question 22

What is the definition of a random variable

Answer 22

If the values assumed by a variable are determined by chance factors. I. E. They cannot be exactly predicted in advance, the variable is called a random variable.

Question 23

How can a random variable be defined as continuous?

Answer 23

A random variable is continuous if it can assume any value within a specified interval of values.

Question 24

What is the definition of the cumulative frequency distribution function or cumulative relative frequency of a sample

Answer 24

Cumulative (frequency) distribution function (cdf) of a sample at x gives the fraction of elements in the sample which are smaller or equal to x.

Question 25

What is the definition of the cumulative distribution function of a random variable

Answer 25

Cumulative distribution function of a random variable x represents the probability that the random variable assumes a value smaller than or equal to x.

Question 26

What is the probability that a continuous random variable assumes a value in the interval between a and b?

Answer 26

The probability that a continuous random variable assumes a value in the (a,b) interval is equal to the area under the curve of the probability distribution function between a and b.

Question 27

Define the variance of a random variable and the variance of a random sample in words

Answer 27

Variance of a random variable if the expected value of the squared deviation of the random variable from its mean.
Variance of a random sample: a statistic estimating the variance of a random variable it population from which the random sample has been taken

Question 28

What is the difference between the standard deviation and standard error of the mean if a sample? Write your answer in words. Not formula.

Answer 28

The SD of a sample gives us an unbiased estimation of the population SD, whereas the SEM is the SD of the sample mean. I. E. It describes how accurately the sample mean approaches the population mean. If the number of elements of the sample increases, the SD approaches the square root of the population variance, the SEM approaches 0.

Question 29

Define the coefficient of variation (CV) in words and with formula

Answer 29

The coefficient of variance (CV) Is the standard deviation expressed as the percentage of the mean.

CV=100 x SD/mean

Question 30

What is an ordered array

Answer 30

An ordered array is a listing of the values of a sample from the smallest to the largest values

Question 31

Define the median of a sample

Answer 31

The median of a sample is the value which divided it into two equal parts such that the number of values equal to or greater than the median is equal to the number of values equal to or less than the median. If the number of elements is odd, the median will be the middle value in the ordered array. If the number of the elements is even 5$3 median will be the average of the two middle values in the ordered array.

Question 32

Define the i-th percentile of a sample!

Answer 32

The i-th percentile of a sample is the smallest score that is equal ticket greater than i% of the observations.

Question 33

Define the first second and third quartile (Q1 Q2 Q3) of a sample.

Answer 33

The first second and third quartile of a sample are the smallest scored which are equal to or greater than 25% 50% and 75% respectively if the observations
Or
Q1 is the 25th percentile Q2 is the 50th percentile(or the median) and Q3 is the 75th percentile.

Question 34

Define the mode of a sample

Answer 34

The mode of a sample is the cue which occurs the most frequently

Question 35

How can a histogram be constructed

Answer 35

The class intervals are displayed in the horizontal axis. Above each class interval a bar is erected doc that the height corresponds to the frequency of the respective class interval

Question 36

Which two properties of the Poisson distribution are equal to 入 , the parameter if the distribution?

Answer 36

Parameter 入 if a Poisson distribution is equal to the mean and the variance of the distribution.

Question 37

When does a random variable follow a standard normal distribution

Answer 37

If it follows a normal distribution and the mean and standard deviation are 0 and 1, respectively