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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
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Statistical estimation
Lesson 6 Statistical estimation
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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).
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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.
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2. Punctual estimation UNBIASED CONSISTENT EFFICIENT SUFFICIENT
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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.
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3.1. Confidence interval for the mean
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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
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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
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3.1. Confidence interval for the mean
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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
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3.1. Confidence interval for the mean
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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.
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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
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3.2. Confidence interval for the proportion
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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.
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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
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