{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

unit 11

# unit 11 - Statistics V3100018.001 UNIT 11 Interval...

This preview shows pages 1–14. Sign up to view the full content.

Statistics – V3100018.001 UNIT 11 – Interval estimation Giuseppe Arbia , Catholic University of the Sacred Hearth, Roma, Italy 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 In statistical inference we distinguish 3 problems 1. Point estimation (unit 10) 2. Confidence intervals (this unit) 3. Hypothesis testing (next unit)
3 1. Point estimation We approximate one unknown parameter with a SINGLE value. We want to find the BEST approximation. E. g. what is the best way of estimating the population mean ? We need to specify the criteria to say what is the BEST WAY The BEST estimator is an estimator which is Unbiased Consistent Most efficient

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
4 2. Confidence intervals We acknowledge that an estimate contains an error and we attach to the estimate a measure that expresses the degree of confidence we have on the result (expressed as a probability). P ( m ! ! " μ " m + ! ) = 1 ! " error Probability of error
5 3. Hypothesis testing Both inductive and deductive. We have some a priori idea and we ask the data to confirm or reject this idea. E. g. We believe that μ=μ 0 and we look at a sample to see how likely is this to be true given the observed sample.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
6 Confidence intervals 1. Confidence intervals around a mean . 2. Confidence intervals around a proportion 3. Confidence intervals around a variance 4. Around regression coefficients
7 Sampling distribu6on of the mean How likely is it that the true value of the mean (that is the popula6on mean) falls within a given range ?

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Back once again to our deductive exercise start Suppose again that we have an urn that contains 6 balls each numbered progressively from 1 to 6, and we draw 4 balls from the urn without replacement. We can draw 15 different samples . 6 4 ! " # \$ % & = 6! 4!2! = 15
It is easy to calculate the probability of an interval including the true population mean. E. g. : P(3 μ 4) = 11/ 15 = 0.7333 11 cases out of 15

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
10 If we can assume that the popula6on is normally distributed with KNOWN variance σ 2 And we further assume that the sample is random (SRS), we know that the sample mean is also normally distributed (at a given sample size n ) as : X ~ N ( μ , ! 2 ) m ~ N μ , ! 2 n ! " # \$ % &
11 So that, standardizing we have: ) , ( N ~ n m 1 0 σ μ

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
12 Let us recall the meaning of : If we could draw all possible samples (of a given dimension n arbitrary but constant in each sample) from a popula6on, and in each sample we calculate the sample mea, then the distribu6on of these means would be normal , with a mean equal to the true popula6on mean and a standard devia.on given by the ra6o between the true standard deviaion of the poula6on and the square root of n . The standard devia6on of the disitribu6on of the means is called STANDARD ERROR of the mean . n , N ~ m σ μ