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Sesión 5
Intervalos de Confianza y
pruebas de Hipótesis
Estadística en las
organizaciones CD4001
Dr. Jorge Ramírez Medina
Estimación de intervalo de la
media de una Población : s
desconocida
• Si no se puede tener un estimado de la
desviación estándar de la población s se utiliza la
desviación estándar s de la muestra para estimar s
.

• En este caso, la estimación del intervalo para m
está basada en la distribución t.

Dr Jorge Ramírez Medina
EGADE Business School
Pruebas de hipótesis
• Una cola
– Cola superior
– Cola inferior
s conocida
s desconocida

• Dos colas

Reject H0

a 

s conocida
s desconocida

Do Not Reject H0

z
Dr Jorge Ramírez Medina
EGADE Business School
Hipótesis nula y
alternativa
 Hypothesis testing can be used to determine whether
a statement about the value of a population parameter
should or should not be rejected.
 The null hypothesis, denoted by H0 , is a tentative
assumption about a population parameter.
 The alternative hypothesis, denoted by Ha, is the
opposite of what is stated in the null hypothesis.
 The alternative hypothesis is what the test is
attempting to establish.
Dr Jorge Ramírez Medina
EGADE Business School
Planteamiento de
Hipótesis
• Testing Research Hypotheses
•

The research hypothesis should be expressed as
the alternative hypothesis.

•

The conclusion that the research hypothesis is true
comes from sample data that contradict the null
hypothesis.

Dr Jorge Ramírez Medina
EGADE Business School
Errores

Dr Jorge Ramírez Medina
EGADE Business School
Type I and Type II Errors

Errores

Population Condition

Conclusion

H0 True
(m < 12)

H0 False
(m > 12)

Accept H0
(Conclude m < 12)

Correct
Decision

Type II Error

Type I Error

Correct
Decision

Reject H0
(Conclude m > 12)

Dr Jorge Ramírez Medina
EGADE Business School
Error Tipo I
 Because hypothesis tests are based on sample data,
we must allow for the possibility of errors.


A Type I error is rejecting H0 when it is true.



The probability of making a Type I error when the
null hypothesis is true as an equality is called the
level of significance.



Applications of hypothesis testing that only control
the Type I error are often called significance tests.

Dr Jorge Ramírez Medina
EGADE Business School
Error Tipo II


A Type II error is accepting H0 when it is false.



It is difficult to control for the probability of making
a Type II error.



Statisticians avoid the risk of making a Type II
error by using “do not reject H0” and not “accept H0”.

Dr Jorge Ramírez Medina
EGADE Business School
Summary of Forms for Null and Alternative
Hypotheses about a Population Mean


The equality part of the hypotheses always appears
in the null hypothesis.

 In general, a hypothesis test about the value of a
population mean m must take one of the following
three forms (where m0 is the hypothesized value of
the population mean).

H 0 : m  m0
H a : m  m0

H 0 : m  m0
H a : m  m0

H 0 : m  m0
H a : m  m0

One-tailed
(lower-tail)

One-tailed
(upper-tail)

Two-tailed

Dr Jorge Ramírez Medina
EGADE Business School
p-Value para la prueba de
Hipótesis de dos colas

 Compute the p-value using the following three steps:

1. Compute the value of the test statistic z.
2. If z is in the upper tail (z > 0), find the area under
the standard normal curve to the right of z.
If z is in the lower tail (z < 0), find the area under
the standard normal curve to the left of z.
3. Double the tail area obtained in step 2 to obtain
the p –value.
 The rejection rule:
Reject H0 if the p-value < a .
Dr Jorge Ramírez Medina
EGADE Business School
p-Value para la prueba de
Hipótesis de dos colas

 The critical values will occur in both the lower and
upper tails of the standard normal curve.
 Use the standard normal probability distribution
table to find za/2 (the z-value with an area of a/2 in
the upper tail of the distribution).


The rejection rule is:
Reject H0 if z < -za/2 or z > za/2.

Dr Jorge Ramírez Medina
EGADE Business School
Pasos de la prueba de
Hipótesis
Step 1. Develop the null and alternative hypotheses.
Step 2. Specify the level of significance a.
Step 3. Collect the sample data and compute the test
statistic.
p-Value Approach
Step 4. Use the value of the test statistic to compute the
p-value.

Step 5. Reject H0 if p-value < a.
Dr Jorge Ramírez Medina
EGADE Business School
Pasos de la prueba de
Hipótesis
Critical Value Approach
Step 4. Use the level of significanceto determine
the critical value and the rejection rule.
Step 5. Use the value of the test statistic and the
rejection
rule to determine whether to reject H0.

Dr Jorge Ramírez Medina
EGADE Business School
Ejemplo: Pasta de
dientes
• Two-Tailed Test About a Population Mean: sKnown
The production line for Glow toothpaste
is designed to fill tubes with a mean weight
of 6 oz. Periodically, a sample of 30 tubes
will be selected in order to check the
filling process.
Quality assurance procedures call for
the continuation of the filling process if the
sample results are consistent with the assumption that
the mean filling weight for the population of toothpaste
tubes is 6 oz.; otherwise the process will be adjusted.
Dr Jorge Ramírez Medina
EGADE Business School
Ejemplo: Pasta de dientes


Two-Tailed Test About a Population Mean: s Known

Assume that a sample of 30 toothpaste
tubes provides a sample mean of 6.1 oz.
The population standard deviation is
believed to be 0.2 oz.
Perform a hypothesis test, at the .03
level of significance, to help determine
whether the filling process should continue
operating or be stopped and corrected.
Dr Jorge Ramírez Medina
EGADE Business School
Prueba de dos colas de µ:
s conocida
 p –Value and Critical Value Approaches
1. Determine the hypotheses.

H0: m6
Ha: m  6

2. Specify the level of significance.

a = .03

3. Compute the value of the test statistic.

x  m  6.1  6  2.74
z
s / n .2 30
0

Dr Jorge Ramírez Medina
EGADE Business School
Prueba de dos colas de µ:
s conocida
 p –Value Approach
4. Compute the p –value.

For z = 2.74, cumulative probability = .9969
p–value = 2(1  .9969) = .0062
5. Determine whether to reject H0.
Because p–value = .0062 < a = .03, we reject H0.

We are at least 97% confident that the mean
filling weight of the toothpaste tubes is not 6 oz.
Dr Jorge Ramírez Medina
EGADE Business School
Prueba de dos colas de µ:
s conocida
 p-Value Approach
1/2
p -value
= .0031

1/2
p -value
= .0031

a/2 =

a/2 =

.015

.015

z
z = -2.74
-za/2 = -2.17
Dr Jorge Ramírez Medina
EGADE Business School

0

za/2 = 2.17

z = 2.74
Prueba de dos colas de µ:
s conocida
 Critical Value Approach
4. Determine the critical value and rejection rule.
For a/2 = .03/2 = .015, z.015 = 2.17
Reject H0 if z < -2.17 or z > 2.17

5. Determine whether to reject H0.
Because 2.47 > 2.17, we reject H0.
We are at least 97% confident that the mean
filling weight of the toothpaste tubes is not 6 oz.
Dr Jorge Ramírez Medina
EGADE Business School
Prueba de dos colas de µ:
s conocida
 Critical Value Approach

Sampling
distribution
of z  x  m

0

s/ n

Reject H0

a/2 = .015
-2.17
Dr Jorge Ramírez Medina
EGADE Business School

Reject H0

Do Not Reject H0

a/2 = .015
0

2.17

z
Prueba de Hipótesis de µ:
s desconocida
• Test Statistic
t

x  m0
s/ n

This test statistic has a t distribution
with n - 1 degrees of freedom.

Dr Jorge Ramírez Medina
EGADE Business School
Prueba de Hipótesis de µ:
s desconocida


Rejection Rule: p -Value Approach
Reject H0 if p –value < a



Rejection Rule: Critical Value Approach
H0: mm
H0: mm

Reject H0 if t > ta

H0: mm

Dr Jorge Ramírez Medina
EGADE Business School

Reject H0 if t < -ta

Reject H0 if t < - ta or t > ta
Examen de la sesión
• Seis preguntas
• Límite de tiempo para contestar el
examen 1hr.
• Puede utilizar computadora, apuntes y
formatos de Excel
• No puede consultar con sus compañeros
• Al iniciar el examen no es permitido
comunicarse electrónicamente ni
personalmente..
Dr. Jorge Ramírez Medina
EGADE Business School
Casos a resolver en Clase
Resuelva en equipos los dos casos para
clase
1. Salarios de inicio para MBA
2. Máquina industrial de relleno de líquido

Las indicaciones se las dará el profesor en
la sesión de clase. Suba sus resultados de
manera individual en la plataforma.
Dr. Jorge Ramírez Medina
EGADE Business School
Asignación para
la siguiente sesión

Dr. Jorge Ramírez Medina
EGADE Business School
Fin Sesión Cinco

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S05 ad4001

  • 1. Sesión 5 Intervalos de Confianza y pruebas de Hipótesis Estadística en las organizaciones CD4001 Dr. Jorge Ramírez Medina
  • 2. Estimación de intervalo de la media de una Población : s desconocida • Si no se puede tener un estimado de la desviación estándar de la población s se utiliza la desviación estándar s de la muestra para estimar s . • En este caso, la estimación del intervalo para m está basada en la distribución t. Dr Jorge Ramírez Medina EGADE Business School
  • 3. Pruebas de hipótesis • Una cola – Cola superior – Cola inferior s conocida s desconocida • Dos colas Reject H0 a  s conocida s desconocida Do Not Reject H0 z Dr Jorge Ramírez Medina EGADE Business School
  • 4. Hipótesis nula y alternativa  Hypothesis testing can be used to determine whether a statement about the value of a population parameter should or should not be rejected.  The null hypothesis, denoted by H0 , is a tentative assumption about a population parameter.  The alternative hypothesis, denoted by Ha, is the opposite of what is stated in the null hypothesis.  The alternative hypothesis is what the test is attempting to establish. Dr Jorge Ramírez Medina EGADE Business School
  • 5. Planteamiento de Hipótesis • Testing Research Hypotheses • The research hypothesis should be expressed as the alternative hypothesis. • The conclusion that the research hypothesis is true comes from sample data that contradict the null hypothesis. Dr Jorge Ramírez Medina EGADE Business School
  • 6. Errores Dr Jorge Ramírez Medina EGADE Business School
  • 7. Type I and Type II Errors Errores Population Condition Conclusion H0 True (m < 12) H0 False (m > 12) Accept H0 (Conclude m < 12) Correct Decision Type II Error Type I Error Correct Decision Reject H0 (Conclude m > 12) Dr Jorge Ramírez Medina EGADE Business School
  • 8. Error Tipo I  Because hypothesis tests are based on sample data, we must allow for the possibility of errors.  A Type I error is rejecting H0 when it is true.  The probability of making a Type I error when the null hypothesis is true as an equality is called the level of significance.  Applications of hypothesis testing that only control the Type I error are often called significance tests. Dr Jorge Ramírez Medina EGADE Business School
  • 9. Error Tipo II  A Type II error is accepting H0 when it is false.  It is difficult to control for the probability of making a Type II error.  Statisticians avoid the risk of making a Type II error by using “do not reject H0” and not “accept H0”. Dr Jorge Ramírez Medina EGADE Business School
  • 10. Summary of Forms for Null and Alternative Hypotheses about a Population Mean  The equality part of the hypotheses always appears in the null hypothesis.  In general, a hypothesis test about the value of a population mean m must take one of the following three forms (where m0 is the hypothesized value of the population mean). H 0 : m  m0 H a : m  m0 H 0 : m  m0 H a : m  m0 H 0 : m  m0 H a : m  m0 One-tailed (lower-tail) One-tailed (upper-tail) Two-tailed Dr Jorge Ramírez Medina EGADE Business School
  • 11. p-Value para la prueba de Hipótesis de dos colas  Compute the p-value using the following three steps: 1. Compute the value of the test statistic z. 2. If z is in the upper tail (z > 0), find the area under the standard normal curve to the right of z. If z is in the lower tail (z < 0), find the area under the standard normal curve to the left of z. 3. Double the tail area obtained in step 2 to obtain the p –value.  The rejection rule: Reject H0 if the p-value < a . Dr Jorge Ramírez Medina EGADE Business School
  • 12. p-Value para la prueba de Hipótesis de dos colas  The critical values will occur in both the lower and upper tails of the standard normal curve.  Use the standard normal probability distribution table to find za/2 (the z-value with an area of a/2 in the upper tail of the distribution).  The rejection rule is: Reject H0 if z < -za/2 or z > za/2. Dr Jorge Ramírez Medina EGADE Business School
  • 13. Pasos de la prueba de Hipótesis Step 1. Develop the null and alternative hypotheses. Step 2. Specify the level of significance a. Step 3. Collect the sample data and compute the test statistic. p-Value Approach Step 4. Use the value of the test statistic to compute the p-value. Step 5. Reject H0 if p-value < a. Dr Jorge Ramírez Medina EGADE Business School
  • 14. Pasos de la prueba de Hipótesis Critical Value Approach Step 4. Use the level of significanceto determine the critical value and the rejection rule. Step 5. Use the value of the test statistic and the rejection rule to determine whether to reject H0. Dr Jorge Ramírez Medina EGADE Business School
  • 15. Ejemplo: Pasta de dientes • Two-Tailed Test About a Population Mean: sKnown The production line for Glow toothpaste is designed to fill tubes with a mean weight of 6 oz. Periodically, a sample of 30 tubes will be selected in order to check the filling process. Quality assurance procedures call for the continuation of the filling process if the sample results are consistent with the assumption that the mean filling weight for the population of toothpaste tubes is 6 oz.; otherwise the process will be adjusted. Dr Jorge Ramírez Medina EGADE Business School
  • 16. Ejemplo: Pasta de dientes  Two-Tailed Test About a Population Mean: s Known Assume that a sample of 30 toothpaste tubes provides a sample mean of 6.1 oz. The population standard deviation is believed to be 0.2 oz. Perform a hypothesis test, at the .03 level of significance, to help determine whether the filling process should continue operating or be stopped and corrected. Dr Jorge Ramírez Medina EGADE Business School
  • 17. Prueba de dos colas de µ: s conocida  p –Value and Critical Value Approaches 1. Determine the hypotheses. H0: m6 Ha: m  6 2. Specify the level of significance. a = .03 3. Compute the value of the test statistic. x  m  6.1  6  2.74 z s / n .2 30 0 Dr Jorge Ramírez Medina EGADE Business School
  • 18. Prueba de dos colas de µ: s conocida  p –Value Approach 4. Compute the p –value. For z = 2.74, cumulative probability = .9969 p–value = 2(1  .9969) = .0062 5. Determine whether to reject H0. Because p–value = .0062 < a = .03, we reject H0. We are at least 97% confident that the mean filling weight of the toothpaste tubes is not 6 oz. Dr Jorge Ramírez Medina EGADE Business School
  • 19. Prueba de dos colas de µ: s conocida  p-Value Approach 1/2 p -value = .0031 1/2 p -value = .0031 a/2 = a/2 = .015 .015 z z = -2.74 -za/2 = -2.17 Dr Jorge Ramírez Medina EGADE Business School 0 za/2 = 2.17 z = 2.74
  • 20. Prueba de dos colas de µ: s conocida  Critical Value Approach 4. Determine the critical value and rejection rule. For a/2 = .03/2 = .015, z.015 = 2.17 Reject H0 if z < -2.17 or z > 2.17 5. Determine whether to reject H0. Because 2.47 > 2.17, we reject H0. We are at least 97% confident that the mean filling weight of the toothpaste tubes is not 6 oz. Dr Jorge Ramírez Medina EGADE Business School
  • 21. Prueba de dos colas de µ: s conocida  Critical Value Approach Sampling distribution of z  x  m 0 s/ n Reject H0 a/2 = .015 -2.17 Dr Jorge Ramírez Medina EGADE Business School Reject H0 Do Not Reject H0 a/2 = .015 0 2.17 z
  • 22. Prueba de Hipótesis de µ: s desconocida • Test Statistic t x  m0 s/ n This test statistic has a t distribution with n - 1 degrees of freedom. Dr Jorge Ramírez Medina EGADE Business School
  • 23. Prueba de Hipótesis de µ: s desconocida  Rejection Rule: p -Value Approach Reject H0 if p –value < a  Rejection Rule: Critical Value Approach H0: mm H0: mm Reject H0 if t > ta H0: mm Dr Jorge Ramírez Medina EGADE Business School Reject H0 if t < -ta Reject H0 if t < - ta or t > ta
  • 24. Examen de la sesión • Seis preguntas • Límite de tiempo para contestar el examen 1hr. • Puede utilizar computadora, apuntes y formatos de Excel • No puede consultar con sus compañeros • Al iniciar el examen no es permitido comunicarse electrónicamente ni personalmente.. Dr. Jorge Ramírez Medina EGADE Business School
  • 25. Casos a resolver en Clase Resuelva en equipos los dos casos para clase 1. Salarios de inicio para MBA 2. Máquina industrial de relleno de líquido Las indicaciones se las dará el profesor en la sesión de clase. Suba sus resultados de manera individual en la plataforma. Dr. Jorge Ramírez Medina EGADE Business School
  • 26. Asignación para la siguiente sesión Dr. Jorge Ramírez Medina EGADE Business School