Sec. 6.3 ESTIMATION OF /i AND <r 2 165 



of 10 observations taken from a normal population has given x = 8 

 and s £ = 2. Then t = (8 — fi)/2, a function of m- Before the sample 

 was taken it could be reasoned that there were 92 chances in 100 that 

 the t to be obtained would have some size between —2 and +2. Like- 

 wise, after the sample is taken, the assumption that the t does lie 

 within these limits runs a risk of 8 in 100 of being wrong as a result of 

 sampling variation. 



What does the assumption that the t obtained from the sample 

 satisfies the inequality (6.35) require of fi now that t = (8 — ju.)/2? 

 The quantity (8 — /*)/2 must be at least as large as —2 but no 

 larger than +2; therefore, (8 — /*) must be at least as large as —4 

 but not larger than +4. It follows that /a must be some number 

 from 4 to 12 unless an 8-in-100 event has occurred. We never actu- 

 ally know in practice if such a t has been got; but we do know that 

 the odds against it are 92:8. 



The probability, .92, associated with the expression (6.35) is called 

 the confidence coefficient for the confidence interval 4 to 12 because it 

 measures the confidence we can put in the inference that p. lies within 

 these limits. This usage is identical with that of Chapter 5. That 

 is, a method for basing decisions on sampling evidence has been 

 presented; and, although we know it is not infallible, we know what 

 risk of error we run when we use the method. 



Obviously, other confidence coefficients besides .92 could be used. 

 For example, 95 and 99 per cent confidence limits are quite common. 

 They require the use of the following inequalities for 9 degrees of 

 freedom: 



x — n 

 — 2.26 < < +2.26 for 95 per cent confidence limits, CI95. 



x — n 

 — 3.25 < < +3.25 for 99 per cent confidence limits, CI99. 



These two inequalities and that of (6.35) can be put into a more con- 

 venient form simply by multiplying through each (all three members) 

 by Sx and then transferring the x to the outer members of the inequali- 

 ties. The final results for 92, 95, and 99 per cent confidence intervals 

 are as follows for 9 degrees of freedom: 



(6.36) (x - 2 Si ) < M < (x + 2s £ ) for a CI 92 ; 



(6.37) (x - 2.26%) < m < (x + 2.26s;,) for a CI 95 ; 



(6.38) (x - 3.25Si) < M < (x + 3.25s*) for a CI 99 . 



