apunts unitat 3, 4 i 5, Apuntes de Estadística

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STATISTICS

Degree in Psychology

Academic year 2013 – 2014

UNITS 3, 4 & 5

PROBABILITY INTERVALS

CONFIDENCE INTERVALS

SAMPLING SIZE CALCULATIONS

Chapter 5 of the book: Peró, M., Leiva, D., Guàrdia, J., & Solanas, A. (coord.) ( 2012 ). Estadística

aplicada a las ciencias sociales mediante R y R-Commander. Madrid: Garceta Grupo Editorial.

Chapter 5 of the book: Guàrdia, J., Freixa, M., Peró, M., & Turbany, J. (2008). Análisis de datos en

psicología. 2ª edición. Madrid: DELTA Publicaciones.

Characteristics of a good estimator

• Not bias:

• Consistent:

• Efficient:

• Sufficient

ˆ E

p   

1 2

Var   Var 

sampling error : SE  

Probability and confidence intervals

• Probability interval : Interval that determines between

which values would be the statistic in samples of size n

fixing the value of . Fixed n &  is unique.

• Confidence Interval: Interval centered in the statistic,

The parameter would be inside this interval with a

probability 1 - . Fixed n &  there would be as many

confidence intervals as samples you obtain.

Probability interval: proportions

• Condition of application

 · n  5

(1 - ) · n  5

• PI: ^ ^ z  / 2 ^  p

 

n

p

  

  

1

 

1

1



 

   N

N n

n

p

  

Example PI for proportions

• The proportion of people greater than 65 years that have

depression in the population is 0,15. ¿Between wath values

would be this proportion in a geriatric center that have 40

residents? (confidence level = 95%)

• Is a probability interval because we have information in the

population and we need the information in the sample:  =

0,15 & p?

• C.A.: 40 · 0,15 = 6 & 40 · (1 - 0,15) = 34  Yes
• PI: 0 , 039 0 , 261 95 %

40

0 , 15 0 , 85 0 , 15 1 , 96   

  CL

Example PI for means

• ¿Between what values would be the age mean in a group of 30

persons greater than 65 years that live in community if in the

population the mean is 72 years and the variance is 5 years^2?

(confidence level = 96%)

• Is a probability interval of the mean, we have the information

in the population and we want to known the sample mean.

• We assume that the age follows the normal distribution in the

population.

• PI: 71 , 16 72 , 84 96 %

30

5 72  2 , 0537    CL 

Confidence Interval: proportions

• CI:

• Condition of application

p  z  / 2   p

 

n

p p p

  

1 

 

1

1



 

   N

N n

n

p p  p

 

ˆ 5  1 ˆ  5

ˆ 5 1 ˆ 5

    

    

n n

n n

s s

i i

 

Confidence interval: means

• Normal distribution
• CI:
  • n large & ^2 known
  • n large & ^2 unknown
  • n small & ^2 knwon
  • n small & ^2 unknown
• If we known N the SE is

corrected by

x  z  / 2   n

 1



N

N n

x  z / 2  S n

x  z  / 2   n

x  t ( , n  1 )  S n

Example CI for means

• In a sample of 20 persons with initial dementia (Reisberg

2. the mean in MEC of Lobo is 24 and the standard

deviation is 4,3 ¿Between what values would be this mean

in the population? (Confidence level = 95%).

• Is a CI of the mean, we have the sample mean (24) and we

want to known the population mean.

• We assume that the values in MEC are normal distributed.
• Small sample and population variance unknown.

21 , 99 26 , 01 95 %

20

4 , 3 24  2 , 093    CL 

Example sample size calculations

• Obtain the sample size

to estimate the

proportion of aged

people that feel alone.

Work with a  of 5% and

an accuracy of 3%.

• Obtain the sample size

to estimate the age

mean of people that

live in residential

centers. Work with a 

of 5% and an accuracy

of 1,6 (s = 6).

2

2



n  54

2

2 2

n 

Example sample size calculations

• Obtain the sample size to

estimate the proportion of

aged people that feel

alone. Work with a  of

5% and an accuracy of 3%.

We known that the

population size is of

250.000 habitants.

• Obtain the sample size to

estimate the age mean of

people that live in

residential centers. Work

with a  of 5% and an

accuracy of 1,6 (s = 6).

We known that the

population size is of

90.000 habitants.

     

1062 , 5798 1063 0 , 03 250000 1 1 , 96 0 , 5 1 0 , 5

250000 1 , 96 0 , 5 1 0 , 5 2 2

2        

    n   

53 , 99 54 1 , 6 90000 1 1 , 96 6

90000 1 , 96 6 2 2 2

2 2      

  n 