Termoquímica nivel bachillerato.
Principales conceptos y ejercicios resueltos
- Principios de la termodinámica
- Ejercicios resueltos
- Entalpías y Energías de reacción
- Espontaneidad de las reacciones químicas
- Entropía
ejercicios basicos para psirometria masa molar volumen molar en mezclas de gases no reaccionantes como los gases ideales y en algunos casos gases reales bajo condiciones de presion y temperatura especifica
Termoquímica nivel bachillerato.
Principales conceptos y ejercicios resueltos
- Principios de la termodinámica
- Ejercicios resueltos
- Entalpías y Energías de reacción
- Espontaneidad de las reacciones químicas
- Entropía
ejercicios basicos para psirometria masa molar volumen molar en mezclas de gases no reaccionantes como los gases ideales y en algunos casos gases reales bajo condiciones de presion y temperatura especifica
OBJECTIVES:
Recognize the differences between categorical data and continuous data
Discuss assumptions of chi square distribution
Correctly interpret and use the terms:
chi-square test of independence,
contingency table
degrees of freedom,
“2x2” and “r x c” table.
Calculate expected numbers of the cells of a contingency table .
Calculate chi-square test statistic and its appropriate degrees of freedom.
Refer the chi-square table to obtain tabulated value.
Categorical variables take on values that are names or labels, such as ethnicity (e.g., Sindhi, Punjabi, Balochi etc.) and methods of teaching (e.g. lecture, discussion, activity based etc.)
Quantitative variables are numerical. They represent a measurable quantity. For example, the number of students taking Biostatistics Supplementary classes .
CHI-SQUARE TEST:
It is used to determine whether there is a significant association between the two categorical variables from a single population.
CHI-SQUARE DISTRIBUTION PROPERTIES:
As the degrees of freedom increases, the chi-square
curve approaches a normal distribution
It has many shapes which are based on its degree of freedom (df)
Distribution is skewed to the right
A chi-square distribution takes positive values only.
Commonly used approaches are:
Test for independence
Test of homogeneity
CHI-SQUARE TEST OF INDEPENDENCE:
A chi-square test of independence is used when we want to see if there is a relationship/association between two categorical variables.
EXAMPLES OF RELATIONSHIPS
BETWEEN QUALITATIVE VARIABLES:
Qualitative variables are either ordinal or nominal.
Examples:
Do the nurses feel differently about a new postoperative procedure than doctors?
Preference (Old/New) Subjects (Nurses/ Doctors)
Is there any relationship between Soya Use & Lung cancer?
Soya Intake (yes/no) Lung cancer (yes/no)
Is there any relationship between parent’s and their children Children’s Education (Illiterate/Up to Intermediate/Graduate)
education?
Parent’s Education (Illiterate/Up to Intermediate/Graduate)
CONTINGENCY TABLE:
The table which classifies categories of the qualitative
variable.
The number of individuals or items assigned to each category is called the frequency.
WHAT INFORMATION DOES CONTINGENCY TABLE REVEAL?
When we consider two categorical variables at a time, then an observation will belong to a particular category of variable one as well as a particular category of variable two. This type of table is referred as contingency table.
The simplest form of contingency table is a 2x2 contingency table with both
variables having exactly two categories.
WHAT OTHER INFORMATION DOES
CONTINGENCY TABLE REVEAL?
In this table Two independent categorical variables that
form a “r x c” contingency table, where “r” is the number of rows (number of categories in first variable e.g. helmet used at the time of accident or not?) and “c” is the number of columns (number of categories in the second variable e.g. got severe brain injury.
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4. Chi-Cuadrado ( 𝜒2) es
el nombre de una
prueba de hipótesis que
determina si dos
variables están
relacionadas o no.
1
2
3
4
5
6
7
Realizar una conjetura.
Escribir la hipótesis
nula y la alternativa.
Calcular el valor de -
Determinar el valor de
p y el grado de libertad.
Obtener el valor
crítico.
Realizar una comparación
entre el chi-cuadrado
calculado y el valor crítico.
Interpretar la
comparación.
Pasos
5. TABLA DE CONTINGENCIA
Es la tabla que contiene los datos
obtenidos contados y organizados.
USO DE CINTURÓN DE
SEGURIDAD
GENERO SI NO
FEMENINO 50 25
MASCULINO 40 45
6. ALTERNATIVA (H1): Es aquella
en la que se asegura que los dos
parámetros analizados sí son
dependientes.
FORMULACIÓN DE HIPÓTESIS
NULA (H0): Es aquella en la que se
asegura que los do parámetros
analizados son independientes uno
del otro.
7. H0: El uso del cinturón de
seguridad es independiente
del género.
El uso de cinturón de seguridad, en los
conductores, está relacionado con el género.
H1: El uso del cinturón de
seguridad no es independiente
del género.
8. Para calcular todos y cada uno de los valores de la tabla de
frecuencias esperadas se realiza:
Variable
1
Variable 2 Total
C1 C2
Fila 1 A11 A12 Tf1
Fila 2 A21 A22 Tf2
Total TC1 TC2 nTotal
𝐹𝑒𝐴11 =
𝑇𝑐1 ∗ 𝑇𝑓1
𝑛𝑇𝑜𝑡𝑎𝑙
𝐹𝑒𝐴12 =
𝑇𝑐2 ∗ 𝑇𝑓1
𝑛𝑇𝑜𝑡𝑎𝑙
𝐹𝑒𝐴21 =
𝑇𝑐1 ∗ 𝑇𝑓2
𝑛𝑇𝑜𝑡𝑎𝑙
𝐹𝑒𝐴22 =
𝑇𝑐2 ∗ 𝑇𝑓2
𝑛𝑇𝑜𝑡𝑎𝑙
Variable
1
Variable 2 Total
C1 C2
Fila 1 FeA11 FeA12 Tf1
Fila 2 FeA21 FeA22 Tf2
Total TC1 TC2 nTotal
TABLA DE FRECUENCIAS ESPERADAS
USO DE CINTURÓN DE
SEGURIDAD
Total
GENERO SI NO
FEMENINO 50 25 75
MASCULINO 40 45 85
Total 90 70 160
Nivel de significancia
2,5%
𝑋2𝑐𝑎𝑙𝑐 =
𝑓0 − 𝑓𝑒
2
𝑓𝑒
9. Ho: No existe relación entre atención y el genero para los pacientes con covid-19, en
Colombia.
Ha: Existe relación (Dependencia) en atención y genero para los pacientes con covid-19
en Colombia.
Atencion
Genero
Total
Masculino Femenino
Casa 47130 50469 97599
Hospital 6900 5052 11952
Total 54030 55521 109551
Atencion
Genero
Total
Masculino Femenino
Casa 48135,33 49463,67 97599,00
Hospital 5894,67 6057,33 11952,00
Total 54030,00 55521,00 109551,00
𝑋2𝑐𝑎𝑙𝑐 =
𝑓0 − 𝑓𝑒
2
𝑓𝑒
Como el valor de Chi-
cuadrado cae en la zona de
rechazo, se acepta la
hipótesis alterna y entonces
se encuentra relación o
dependencia entre las
variables atención y genero
para los pacientes con covid-
19 en Colombia.
fo fe (fo- fe)2 (fo- fe)2/fe
47130 48135,33 1010688,41 20,9968106
6900 5894,67 1010688,41 171,458014
50469 49463,67 1010688,41 20,4329442
5052 6057,33 1010688,41 166,85378
379,741549
10. Ho: El uso del cinturón de seguridad no depende del genero
Ha: El uso del cinturón de seguridad depende del genero
Fo Fe (Fe-Fo)2 (Fe-Fo)2/Fe
50 42,1875 61,0351563 1,44675926
40 47,8125 61,0351563 1,27655229
25 32,8125 61,0351563 1,86011905
45 37,1875 61,0351563 1,64128151
6,22471211
Gl1,2,5=5.0239
Como el Chicuadrado cae en la
zona de rechazo se acepta la
hipótesis alternativa y entonces
se encuentra una relación
enter el genero y el usar el
cinturón de seguridad.
11. CHI – CUADRADO CALCULADO
Para obtener el valor de Chi-
Cuadrado Calculado se tiene la
fórmula
GRADO DE LIBERTAD v
Para calcular el grado de libertad (v) se realiza:
v =(Cantidad de filas−1)(Cantidad decolumnas−1)
𝑋2
𝑐𝑎𝑙𝑐 =
𝑓0 − 𝑓𝑒
2
𝑓𝑒
𝑓0: Frecuencia del
valor observado
𝑓𝑒: Frecuencia del
valor esperado
12. NIVEL DE SIGNIFICANCIA
Por lo general se trabaja con
un nivel de significancia de
0.05, que indica que hay
una probabilidad del 0.95
de que la hipótesis nula sea
verdadera.
Es el error que se
puede cometer al
rechazar la
hipótesis nula
siendo verdadera.
VALORES CRÍTICOS DE LA
DISTRIBUICIÓN
𝑷 = 𝒑(𝒙 ≤ 𝒄)
13. Frecuencias Observadas
Atencion
Genero
Total
fo fe (fo-fe)2 (fo-fe)2/fe
Masculino Femenino 47130 49479,296 5519191,7 111,545477
Casa 47130 50469 97599 6900 6059,248 706863,926 116,658689
Hospital 6900 5052 11952 281802 280293,455 2275708,02 8,11901946
Recuperad
o 281802 271083 552885 50469 48119,703 5519196,39 114,697225
Total 335832 326604 662436 5052 5892,751 706862,244 119,954541
271083 272591,545 2275708,02 8,34841747
Frecuencias esperadas Total 479,323369
Atencion
Genero
Total
Masculino Femenino
Casa
49479,296
7
48119,703
3 97599
Hospital
6059,2480
8
5892,7519
2 11952
Recuperad
o
280293,45
5
272591,54
5 552885
Total 335832 326604 662436
Gl 2
Ho: No existe Existe una relación hay
independencia entre la atención y el
genero de los pacientes enfermos de
covid-19
Ha: Existe una relación o dependencia
entre la atención y el genero de los
pacientes enfermos de covid-19
Como el valor del chicuadrado
calculado cae en la zona de rechazo, se
acepta la hipótesis alterna y entonces
existe una dependencia entre la
atención y el genero de los pacientes
enfermos con covid-19.
14. COMPARACIÓN ENTRE LOS VALORES DEL CHI-CUADRADO
CALCULADO Y EL CRÍTICO
Si el valor del chi-
cuadrado calculado es
menor o igual que el chi-
cuadrado crítico entonces
se acepta la hipótesis
nula, caso contrario no se
la acepta.
𝑋2𝑐𝑎𝑙𝑐 ≤ 𝑉𝑎𝑙𝑜𝑟 𝑐𝑟í𝑡𝑖𝑐𝑜
6.2248 ≤ 6.635
15. EJEMPLO
Para estudiar la dependencia entre la práctica de
algún deporte y la depresión, se seleccionó una
muestra aleatoria simple de 100 jóvenes, con los
siguientes resultados:
Sin depresión Con depresión
Deportista 38 9
No deportista 31 22
Determinar si existe
independencia entre la
actividad del sujeto y su
estado de ánimo. Nivel de
significación (5%)
16. fo fe (fo-fe)2 (fo-fe)2/fe
38 32,43 31,0249 0,956672834
31 36,57 31,0249 0,848370249
9 14,57 31,0249 2,129368566
22 16,43 31,0249 1,888307973
5,822719621
5% de significacnia el Chi-
cuadrado es de 3,8415
Ho: No existe una relación entre hacer
deporte y sentir depresión
Ha: Existe una relación entre hacer deporte y
la depresión.
Como el chicuadrado cae en la zona de rechazo se acepta la
hipótesis alterna y entonces existe una relación entre hacer
deporte y sufrir depresión.
17. Practica
Estado de animo
Sin Depresion Con depresion
Deportista 38 9
No deportista 31 22
Ha: El estado de animo de
una persona depende de la
practica de un deporte.
Grados de libertad: (2-1)(2-1)=1*1=1
𝜒 2
=
𝑖=1
𝑛
(𝑓𝑜 − 𝑓𝑒)2
𝑓𝑒
= 5,822
𝜒5%
2
=3,8415
Practica
Estado de animo
Sin Depresion Con depresion
Deportista 32,43 14,57
No deportista 36,57 16,43
fo fe (fo-fe)2 (fo-fe)2/fe
38 32,43 31,0249 0,95667283
9 14,57 31,0249 2,12936857
31 36,57 31,0249 0,84837025
22 16,43 31,0249 1,88830797
5,82271962
Conclusión: El estado de
animo de una persona
Si, depende de la practica de
un deporte.
𝜒 2 cae en la zona de
rechazo acepto alternatica
Ho: estado de animo de las
personas es independiente de La
practica de un deporte.
SOLUCIÓN
18. En un determinado hospital se están realizando diversos estudios comparativos, con el objetivo de
estudiar el número pacientes que llegan, durante una semana al hospital, para ser diagnosticado y el
número de enfermos con un tipo de carcinoma que reciben una determinada terapia. Para ello se
dispone de la siguiente información:
Sexo H H H H H M H M H M H H M M M M H H M M
Terapia NO SI NO SI NO SI NO SI SI SI NO SI NO SI SI NO SI NO SI NO
𝜒 2 =
𝑖=1
𝑛
(𝑓𝑜 − 𝑓𝑒)2
𝑓𝑒
= 0,89
SI NO Total
Hombre 5 6 11
Mujer 6 3 9
Total 11 9 20
SI NO total
Hombre 6,1 5 11
Mujer 5 4,1 9
Total 11 9
Grados de libertad: (2-1)(2-1)=1*1=1 𝜒5%
2
=3,8415
Ha: El genero depende del numero
de terapias tomadas en el hospital
EJEMPLO
Ho: El genero es independiente
del numero de terapias tomadas
en el hospital
19. fo fe (fe-fo)2 (fe-fo)2/fe
6 6 0 0
6 3 9 3
6,857 5,143 2,937796 0,57122224
5,143 3,857 1,653796 0,42877781
4,00000005
Ho: El genero es independiente del numero de terapias
tomadas en el hospital
Ha: El genero depende del numero de terapias tomadas
en el hospital
Como el chi cuadrado calculado cae
en la zona de rechazo, entonces se
acepta la hipótesis alterna y se
determina que el genero depende
del numero de terapias tomadas en
el hospital.
20.
21. En la dócima de McNemar,
los resultados se presentan
en una tabla 2*2 en la
siguiente forma:
En el cuadro de entrada de
frecuencias se presenta al
primero y al segundo
conjunto de respuestas de
los mismos individuos.
En la tabla se usan + y –
para simbolizar respuestas
diferentes.
Clasificación Yi
(+)
𝑌𝑖 = 0
(-)
𝑌𝑖 = 1
Clasificación
X
i
(+)
𝑋𝑖 = 0
A
(0,0)
B
(0,1)
(-)
𝑋𝑖 = 1
C
(1,0)
D
(1,1)
22. Sea: B+C=n
Si n≥20 entonces la
estadística de contraste es:
𝑇1 =
(𝐵 − 𝐶)2
𝐵 + 𝐶
≈ 𝑋(1)
2
𝑇1 =
( 𝐵 − 𝐶 − 1)2
𝐵 + 𝐶
≈ 𝑋(1)
2
La aproximación muestral a la distribución chicuadrada llega a
ser muy buena si se hace una corrección por continuidad
Esto debido a que se usa una distribución continua para
aproximar unas distribución discreta.
Cuando las frecuencias esperadas con pequeñas esa
aproximación puede ser muy pobre, por lo que aplicamos la
corrección “Yates”, siendo el estadístico corregido.
23. REGLA DE DECISIÓN:
Para esta estadística de
contraste, rechazamos H0,
si…
𝑇1 ≥ 𝑋∝/2(𝑙𝑔.𝑙.)
2
𝑇1 ≥ 𝑋1/2(𝑙𝑔.𝑙.)
2
24. Antes Despues Total
Democrata 84 21 105
Republicano 16 4 20
total 100 25 125
A=84
B=21
C=16
D=4
DATOS B+C=37
𝑇1 =
𝐵−𝑐 2
𝐵+𝐶
=
(5)2
37
=25/37=0,67
𝜒5%
2
=3,8415
O,67<3,8415 Se acepta la
hipótesis alterna
Ho: La forma de pensar de
los candidatos antes del
debate no cambia
Ha: La forma de pensar de los
candidatos después del debate
si cambia.
EJEMPLO
25. T1 =
𝐵−𝑐 2
𝐵+𝐶
=
(5)2
37
=25/37=0,67
Ho: La forma de pensar de los candidatos antes del
debate no cambia.
Ha: La forma de pensar de los candidatos después del
debate si cambia.