1. Design and Data Analysis in Psychology I
School of Psychology Dpt. Experimental Psychology
Design and Data Analysis in Psychology I
Salvador Chacón Moscoso
Susana Sanduvete Chaves

2. Statistical estimation
Lesson 6
Statistical estimation

3. 1. Introduction
Parameters (values in the population) are not usually known.
Statistics can be calculated from samples.
A statistic obtained in a randomized sample will take a concrete value from all the possible values.
A parameter obtained in a population is only one value; it is a constant.
The value that a statistic takes in a randomized sample is a estimation of the parameter (e.g., the statistic is a estimator of μ; the statistic S2 is a estimator of σ2).

4. 2. Punctual estimation
It consist in obtaining a concrete value for a parameter when we infer it (e.g., μ=10).
It can be done when the sample is representative of the population:
Randomized selection.
Adequate sample size.
A statistic is a good estimator of its parameter when is:
Unbiased: its expected value is equal to its parameter.
Consistent: its variance is 0 when n tends to ∞.
Efficient: the lowest standard error.
Sufficient: its based on all the information of the sample.

5. 2. Punctual estimation
UNBIASED
CONSISTENT
EFFICIENT
SUFFICIENT

6. 3. Interval estimation
It consists in obtaining a range of values with a high probability of containing the parameter (usually 95%; 99%).
Elements:
Risk level = α Level of confidence = 1-α.
Standard score: Zα/2.
Standard error:
Maximum error:
Confidence limits:
Upper.
Lower.

7. 3.1. Confidence interval for the mean

8. 3.1. Confidence interval for the mean
Most common values:
α = α = Z α/2 = 1.96
α = α = Z α/2 = 2.58
Risk level: α/2
Level of confidence
1-α
Risk level: α/2

9. 3.1. Confidence interval for the mean
0.95
α/2=0.025
α/2=0.025
0.475
0.475
Z=-1.96
Z=+1.96
α =0.05

10. 3.1. Confidence interval for the mean

11. 3.1. Confidence interval for the mean
0.99
α/2=0.005
α/2=0.005
0.495
0.495
Z=-2.58
Z=+2.58
α =0.01

12. 3.1. Confidence interval for the mean

13. 3.1. Confidence interval for the mean. Example
We obtain, randomly, a sample of 101 participants. Their mean in a test is 80 and the standard deviation is 10. Calculate the limits in which the true mean of the population is, considering a level of confidence of 0.99.

14. 3.1. Confidence interval for the mean. Example
The true mean in the test
will be between and
82.58, with a level of
confidence of 0.99

15. 3.2. Confidence interval for the proportion

16. 3.2. Confidence interval for the proportion. Example
A politician would like to know if a group of workers would accept a new work law. In order to know it, he asked 10 workers and only 3 did not agree. Calculate the limits in which the true proportion of workers that did not agree is, considering a level of confidence of 0.99.

17. 3.2. Confidence interval for the proportion. Example
The true proportion of
workers that did not agree
is between 0 (because we can not obtain negative values in proportions) and
0.674, with a level of
confidence of 0.99