{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture 23-3 - C.I. 2nd copy

# Lecture 23-3 - C.I. 2nd copy - 1 EXAMPLES OF CONFIDENCE...

This preview shows pages 1–3. Sign up to view the full content.

1 EXAMPLES OF CONFIDENCE INTERVALS Lecture 23 ORIE3500/5500 Summer2011 Li Two sided conﬁdence intervals These are of the form I ( X 1 ,...,X n ) = ( ˆ θ l , ˆ θ u ) , where ˆ θ l and ˆ θ u are two statistics, denoting the lower and upper bounds of the interval. One uses two sided conﬁdence intervals when there is no strong preference to bound the parameter of interest on a particular side, but, a general interest to know where θ is. 1 Examples of Conﬁdence Intervals The crux of the story is that in order to ﬁnd a conﬁdence interval for any parameter, ﬁnd a reasonable estimator for the parameter. Modify this esti- mator (rescale or relocate, usually involving the parameter) to get a statistic with a distribution without the parameter and then ﬁnd the upper and lower bound of the parameter using the quantiles of the distribution. Let us see the examples. Example ( σ is known, CI for μ ) Suppose X 1 ,...,X n are i.i.d. N ( μ,σ 2 ) random vari- ables and we want ﬁnd a 1 - α level conﬁdence interval for μ . As the ﬁrst step, we need to construct a statistic. When σ is known, we will use ¯ X - μ σ/ n . This is because it follows standard normal. At the second step, we ﬁnd the two sided conﬁdence interval of level 1 - α . Write P Φ - 1 ( α/ 2) < ¯ X - μ σ/ n < Φ - 1 (1 - α/ 2) · = 1 - α. (1) 1

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

View Full Document
1 EXAMPLES OF CONFIDENCE INTERVALS By solving Φ - 1 ( α/ 2) < ¯ X - μ σ/ n < Φ - 1 (1 - α/ 2), we conclude that the 2-sided conﬁdence interval for μ is ¯ X - σ Φ - 1 (1 - α/ 2) n , ¯ X - σ Φ - 1 ( α/ 2) n · . (2) Denote z α as the 1 - α quantiles of standard normal, i.e.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 5

Lecture 23-3 - C.I. 2nd copy - 1 EXAMPLES OF CONFIDENCE...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online