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1. Sesión 5
Intervalos de Confianza y
pruebas de Hipótesis
Estadística en las organizaciones
AD4001
Dr. Jorge Ramírez Medina
2. Conceptos de la sesión anterior
• Estadística Inferencial
• Distribución de probabilidad
• Valores Z
• Chebyshev y regla empírica
• Teorema del límite central
• 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
s conocida
s desconocida
Reject H0
a 1
Do Not Reject H0
Dr Jorge Ramírez Medina
EGADE Business School
z
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
7. Type I and Type II Errors
Population Condition
Correct
Decision
Type II Error
Correct
Decision
Type I Error
Accept H0
(Conclude m < 12)
Reject H0
(Conclude m > 12)
H0 True
(m < 12)
H0 False
Conclusion (m > 12)
Dr Jorge Ramírez Medina
EGADE Business School
Errores
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
H0 : m m0
0 : a H m m
One-tailed
(lower-tail)
One-tailed
(upper-tail)
Two-tailed
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).
Dr Jorge Ramírez Medina
EGADE Business School
0 0 H : m m
0 : a H m m
0 0 H : m m
0 : a H m m
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: sKnown
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: 6 m
2. Specify the level of significance.
a = .03
3. Compute the value of the test statistic.
Dr Jorge Ramírez Medina
EGADE Business School
2.74
6.1 6
.2 30
m
n
z x
/
0
s
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. a/2 =
.015
Prueba de dos colas de μ:
z = -2.74 0
z = 2.74
za/2 = 2.17
z
p-Value Approach
a/2 =
.015
-za/2 = -2.17
1/2
p -value
= .0031
1/2
p -value
= .0031
s conocida
Dr Jorge Ramírez Medina
EGADE Business School
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
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
Reject H0 if z < -2.17 or z > 2.17
21. Prueba de dos colas de μ:
s conocida
Critical Value Approach
Dr Jorge Ramírez Medina
EGADE Business School
m
Do Not Reject H0 Reject H0
a/2 = .015
0 2.17
z
Reject H0
-2.17
Sampling
distribution
of
a/2 = .015
n
z x
/
0
s
22. Prueba de Hipótesis de μ:
s desconocida
• Test Statistic
t
x
s n
m 0
/
This test statistic has a t distribution
Dr Jorge Ramírez Medina
EGADE Business School
with n - 1 degrees of freedom.
23. Prueba de Hipótesis de μ:
s desconocida
Rejection Rule: p -Value Approach
Reject H0 if p –value < a
Rejection Rule: Critical Value Approach
Reject H0 if t < -ta
H0: m m Reject H0 if t > ta
Reject H0 if t < - ta or t > ta
H0: m m
H0: mm
Dr Jorge Ramírez Medina
EGADE Business School
24. Ejemplo: Los federales
A State Highway Patrol periodically samples
vehicle speeds at various locations
on a particular roadway.
The sample of vehicle speeds
is used to test the hypothesis
H0: m < 65
The locations where H0 is rejected are deemed
the best locations for radar traps.
Dr Jorge Ramírez Medina
EGADE Business School
25. Ejemplo: Los federales
At Location F, a sample of 64 vehicles shows a
mean speed of 66.2 mph with a
standard deviation of
4.2 mph. Use a = .05 to
test the hypothesis.
Dr Jorge Ramírez Medina
EGADE Business School
26. Prueba de una cola de μ:
s desconocida
1. Determine the hypotheses.
H0: m < 65
Ha: m > 65
2. Specify the level of significance.
a = .05
3. Compute the value of the test statistic.
0 66.2 65
m
2.286
x
/ 4.2/ 64
t
s n
Dr Jorge Ramírez Medina
EGADE Business School
27. Prueba de una cola de μ:
s desconocida
p –Value Approach
4. Compute the p –value.
For t = 2.286, the p–value must be less than .025
(for t = 1.998) and greater than .01 (for t = 2.387).
.01 < p–value < .025
5. Determine whether to reject H0.
Because p–value < a = .05, we reject H0.
We are at least 95% confident that the mean speed
of vehicles at Location F is greater than 65 mph.
Dr Jorge Ramírez Medina
EGADE Business School
28. Prueba de una cola de μ:
s desconocida
Critical Value Approach
4. Determine the critical value and rejection rule.
For a = .05 and d.f. = 64 – 1 = 63, t.05 = 1.669
Reject H0 if t > 1.669
5. Determine whether to reject H0.
Because 2.286 > 1.669, we reject H0.
We are at least 95% confident that the mean speed
of vehicles at Location F is greater than 65 mph.
Location F is a good candidate for a radar trap.
Dr Jorge Ramírez Medina
EGADE Business School
29. a
0 ta =
1.669
Reject H0
Do Not Reject H0
t
Prueba de una cola de μ:
s desconocida
Dr Jorge Ramírez Medina
EGADE Business School
30. 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
31. Caso a resolver
Resuelva en equipos el caso para clase
1. Salarios de inicio para MBA
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
32. Asignación para
la siguiente sesión
Dr. Jorge Ramírez Medina
EGADE Business School
