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**Unformatted text preview: **STS 2610: Quantitative Methods Spring 2014 man“ Id MH’ Estimating [,1 en 6 is Unknown If the variable x is N01, 0'), then
J? is N(,u,0‘/‘/;) and is N(0, l) . This is why we use 2 critical values in the formulas for the 2- §_#
a/J; If a is Unknown: error margin and confidence interval. pllrli (ﬁlma’cr o! 0" , the sample standard deviation. ﬁe. - :y 1-] Must estimate from data using 3 = l
n — Problem: if the variable x is N(/.1,0'), then 3 is N(,u,a/~/;) but x _ ,u does not follow a N(0,l) distribution. 2:
s/JI—i z cricital values should not be
interval. In fact, the correct confidence level will be I used. They will not give use the correct confidence level for our
ower than what we are stating. Solution: if the variable X is N(/1,a), then is Now/Jr?) but x - follows a t distribution with n - 1 degrees of freedom. (in) p
5/5
Facts About t Distributions
samplrsm - I (”3 tribution with n — 1 degrees of freedom I: If the variable T follows a t dis 0 E(T)=O o The density curve is symmetric.
o Varl‘l') > VarlZ) = 1 (Anyt density curve is flatter than the 2 density curve.) As n—boo the t density curves look more and more like the 2 density curve. t an" df= 0-1 NW) owl El Student’ 5 t Distribution i Notortmzasn’mcreases Huntsman: are belt. . antisymmetric. but
have law toils that the
normal > Estimatingthe Mean it of a Normal Population Distribution (0 Untmwn
HOG-am Enos Massin:
C 3 t‘lz. 5‘:- l00£l - aﬁtonﬂdence interval: ‘ + t .5... d‘$® x ‘ é; ‘5 ,e = ‘15
wugmeTableoftCriticalValues ’ l” .65 c"Ia ‘ -0&5 - 0‘ : 8
0‘ t . 63
‘ I ' f/a : u 0‘
mm‘. WhentoUsetProceduresnPractice {”m‘”
“(n namatm . n < :5 Use t procedures only if population distribution has bell shape
/""‘"""" 0 I5 5 n < 30 Use t procedures if the data are roughty symmetric with no outliers.
o 30 s n < 50 Use t procedures unless data are highly skewed 0: contain outliers.
O n > 50 Use t procedures without resetvation. . ...

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