This document covers interval estimation and hypothesis testing. It discusses estimating the population mean and interval when the population standard deviation is known and unknown. Hypothesis testing concepts covered include null and alternative hypotheses, types of errors, test statistics, p-values, and critical values. Examples are provided for one-tailed and two-tailed hypothesis tests of a population mean with the population standard deviation both known and unknown. Students will complete an exam and group assignments analyzing hypothesis testing case studies in the next class.

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

27. Fin Sesión Cinco