Esercitazioni Data Analysis

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VEROg) Values higher than the mean for the first character are associated on average with values higher than the mean for the second character

h) There is an error because can’t be outside the interval [-1; +1] NO

xyi) On average as one of the character increases the other increases and viceversa NO

31) In a population we have two quantitative characters, X and Y. We know that = -1. What can we say?

xya) There is a perfect quadratic relation between X and Y NO

b) = -xy

c) = 0xy

d) = - NOxy

e) The scatter plot is composed of points that are all perfectly aligned NO

f) Correlation in the population can’t be negative

g) As X increases then Y increases and viceversa NO

h) = -xxyy x

32) In a population of 6 persons the average age is 27,2 and the deviance is 45,9. Compute the CV (%) of age in the population.

CV= (dev.standard x media) / 100

Deviance = (x – media) x n2i i

Standard deviation = (x – media) x n ] / n2[( i i

Standard deviation =

2,8√(45,9/6)CV = (2,8 x 27,2) / 100 = 0,833) Given the distribution of the relative frequencies for the character "number of siblings" for a population of a small town in Italy

#sib	Rel. freq.	ln(f)
0	0,01	-4,605
1	0,44	-0,821
2	0,55	-0,598
Tot	1

Select all the answers that you deem correct. a) The relative entropy is equal to 1,0621 b) The relative entropy is equal to 0,6701 (VERO) c) To have maximum entropy the relative frequencies should be all equal to 0,333 (VERO) d) The average number of siblings is equal to 1,54 e) The relative entropy is equal to 5,4833 f) The average for the number of siblings is equal to 1 (VERO) g) The CDF for 1 number of siblings is equal to 0,45 (VERO) (CDF=cumulative distribution funct) h) The total population of the small town in Italy is equal to 1 family 34) Given the distribution of 25 persons by the number of mobiles

#mob	Freq	ECDF
0	7	0,28
1	6	0,52
2	7	0,83
3	5	F 1 (freq relativa di 3: 5/25= 0,2) ECDF di 3 = 0,8+0,2 =
Tot	25

Select all the answers that you

a) The average is 4,50

b) The relative frequency for 2 mobiles is 0,80

c) The cumulative relative frequency for 3 mobiles is equal to 1 VERO

d) The average number of mobiles is 2,12

e) The relative frequency for 2 mobiles is 0,28 VERO

f) The average number of mobiles is 4,50

g) The total amount of the character is 53,0035

5) Given the distribution of 18 persons by the number of mobiles

#mob Freq ECDF

0 8 0,444

1 2 0,556

2 1 0,611

3 7 F 14

Tot 18

a) The average number of mobiles is 2,50

b) The relative frequency for 2 mobiles is 0,06 VERO (1/18= 0,06)

c) The average is 4,50

d) The relative frequency for 2 mobiles is 0,61

e) The average number of mobiles is 1,94

f) The cumulative relative frequency for 3 mobiles is equal to 1 VERO

g) The total amount of the character is 3536

6) In the United State of America, you have been appointed chief statistician consultant for the President. He asks you to draw a random sample of America citizens to get information on the approval rate for his decisions.

Please select all the strategies that you consider appropriate.

1. You set up desks in the streets of New York, Los Angeles and Austin and randomly select people on the street
2. You select all the state capitals and then randomly select citizens from those capitals
3. You set up desks at a random selection of malls in the US and you interview people at random there
4. You randomly select a number of states and then randomly select citizens from those states VERO
5. You ask for the list of all American citizens and randomly select people from that list VERO
6. You randomly select a sample of citizens from the phone directory
7. You set up a survey on the internet and ask the President to go on TV to invite people to fill out the survey
8. You randomly select a sample of states from the 50 states and interview all citizens from those states.

37) Consider the following formula: Select all the statements that you deem correct

1. It has the same unit of measurement of the character x VERO
2. It is a number

1. It is the formula for a variance of a character
2. It is the formula for standard deviation of a character VERO
3. It is the variance for a frequency distribution of a character
4. It is the standard deviation for a frequency distribution of a character VERO

38) The outcome of an experiment is an event. Probability is interested in assessing the likelihood of a particular event happening and it gives each outcome a measurement on a scale from zero to one, where the first is associated with the impossible event and the second with the certain event.

39) A convenient sample:

1. It can be a snowball sample VERO
2. It can be used for inference purposes
3. It used when is no list available of all the units in the population VERO
4. It can be used to gain information on the whole population
5. Can be used when a random sample is not an option VERO
6. Is a reliable and convenient source to predict the behavior of people during election day
7. Is a census survey method

random sample

1. In a contingency table we have consider the characters gender and race. Out of 54 persons we have 2 women and the total black of people is 5. Whats is the expected number of black women in case of independence between the two characters?

GENDER	RACE
Women	2
Men	52
Tot.	54

Expected number of black women = (5 x 2) / 54 = 0.2

41. In the average grade over 4 exams is 24.6 and you gain a mark of 18 at a new exam, what is the overall average grade?

24.6 x 4 = 98.4

98.4 + 18 = 116.4

Overall average grade = 116.4 / 5 = 23.28

3.95

0.50

106.40

17.80

6.4 + (1.5 x 7.6) = 17.80

43. Given the distribution of the relative frequencies for the character number of siblings for a population of a small town in Italy

#sib	Rel Freq	ln(fi)
0	0.23	-1.47
1	0.35	-1.05
2	0.42	-0.868
Tot	1

Relative entropy = -[(0.23 x -1.47) + (0.35 x -1.05) + (0.42 x -0.868)] / ln3

The relative entropy is equal to 0.9741

The total population of the small town in Italy is equal to 1 family

The CDF for 1 number of siblings is equal to 0,58 VERO

The average for the number of siblings is equal to 1 VERO

To have maximum entropy the relative frequencies should be all equal to 0,333 VERO

The relative entropy is equal to 3,083944

In a population the variance for age is 383,1. Considering that the average age is 5,2 compute the coefficient of variation as a percentage.

CV = (stand. Deviation / media) x 100

Standard deviation = = 19,5729383,1

CV = (19,6 / 5,2) x 100 = 376,4

In a population, we have computed- the variance for age which is 30,4- the variance for weight which is 19,2- the covariance between the two characters which is -1,5

Compute the correlation coefficient.

Correlation coefficient = Covarianza / Dev. Standard (X) x Dev. Standard (Y)

Stan dev age = = 5,513630,4

Stan dev weight = = 4,381719,2

Correlation coeff. = -1,5 / (5,5136 x 4,3817) = -0,0646

In a contingency table we have consider

The characters gender and race. Out of 46 persons we have 18 women, and the total number of black people is 11. What is the expected number of black men in case of independence between the two characters?

GENDER RACE

Women 18

Black 11 (11 x 28) / 46 = 6.7

Men 28

White 35

Tot 46

47) The quantity GINI INDEX

a) if the variance is zero then it is zero as well

b) measure of heterogeneity VERO

c) an index with the same unit of measure of the character in the sample

d) an index with the same unit of measure of the character in the population

e) a quantity that is zero if all characters have the same value (Vero?)

f) measure of homogeneity

g) an index that decreases as the units tend to show more and more different values

h) is one if there is minimum homogeneity VERO

48) In a population of 4 persons the ages are 23, 25, 15 and 27. Compute the variance for age.

VARIANZA = [(x - media) x n ] / n2

i i

Media = (23+25+15+27) / 4 = 22.5

Varianza = [(23-22.5) x 1 + (25-22.5) x 1 + (15-22.5) x 1 + (27-22.5) x 1] / 4 = 2

2 2 220,7549) Given the distribution of 27 person by the number of mobiles#mob freq ECDF

0 3 0,1111

9 0,4442

8 0,741

Freq relat di 2 = 8/27 = 0,303

7 F 7/27 = 0,26 + 0,741 = 14TOT 27

a) The average number of mobiles is 6,00

b) The total amount of the character is 70,00

c) The cumulative relative frequency for 3 mobiles is equal to 1VERO

d) The average number of mobiles is 2,59

e) The average is 4,50

f) The relative frequency for 2 mobiles is 0,30 VERO

g) The relative frequency for 2 mobiles is 0,7450)

a) n is the total number of units with X = x and Y = y VERO

ij i i

b) it is a contingency table for characters x and y VERO

c) n is the total number of units that show Y = y VERO.

j jd

n = n * nij i. .j

e) it is a contingency table for characters n and ni, .j

f) n it the total number of units with X = x and Y = yij j i

51) In a population we have computed- the variance for age which is 9,9- the variance for weight which is 51,8- the covariance between the two characters which is -1,3Compute the

Il coefficiente di correlazione è calcolato come la covarianza divisa per il prodotto delle deviazioni standard di X e Y. Nel nostro caso, la deviazione standard dell'età è 3,15 e la deviazione standard del peso è 7,20. Quindi il coefficiente di correlazione è -1,3 / (3,15 x 7,20) = -0,065. In una tabella di contingenza, consideriamo i caratteri di genere e razza. Su un totale di 44 persone, abbiamo 11 donne e il numero totale di persone di colore è 12. Qual è il numero atteso di uomini di colore nel caso di indipendenza tra i due caratteri? (arrotondato a una cifra decimale) GENERE RAZZA Donne 11 Neri 12 (12 x 33) / 44 = 9,0 Uomini 33 Bianchi 32 Tot 44 Dato il numero di auto possedute in un campione: #auto Frequenza 1 63 2 7 3 9 4 12 Tot n 28 Calcoliamo la media aritmetica: MEDIA ARITMETICA = [(1x6)+(3x7)+(10x9)+(12x6)] / 28 = 6,85 In una classe di 154 persone, 23 indossano occhiali. Lavori per strada e incontri una di queste persone. Qual è la probabilità che non indossi occhiali? 154 - 23 = 131 persone che non indossano occhiali P(1 no glass) = 1 / 131 = 0,015 La quantità ENTROPIA:

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