# 2 the sample is small n 30 3 the value of the

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2)The sample issmall(n30).3)The value of the population standarddeviationis unknown.4)The sample values come from a populationwith a distribution that is approximatelynormal.
Test Statisticfor a Studentt-distributionCritical ValuesFound in Table A-3Degrees of freedom (df) =n-1Criticaltvalues to the left of the mean arenegativet=xxsn
Important Properties of theStudenttDistribution1. The Studenttdistribution is different for different sample sizes (seeFigure 6-5 in Section 6-3).2. The Studenttdistribution has the same general bell shape as thenormal distribution; its wider shape reflects the greater variability thatis expected with small samples.3. The Studenttdistribution has a mean oft= 0 (just as the standardnormal distribution has a mean ofz= 0).4. The standard deviation of the Studenttdistribution varies with thesample size and is greater than 1 (unlike the standard normaldistribution, which has a= 1).5. As the sample sizengets larger, the Studenttdistribution get closer tothe normal distribution.For values ofn> 30, the differences are sosmall that we can use the critical z values instead of developing a muchlarger table of criticaltvalues.(The values in the bottom row of TableA-3 are equal to the corresponding criticalzvalues from the normaldistributions.)
Figure 7-11Choosing between the Normal and Studentt-Distributions when Testing a Claim about a Population MeanµIsn > 30?Is thedistribution ofthe population essentiallynormal ? (Use ahistogram.)NoYesYesNoNoIsknown?Usenormal distribution withx - µx/nZ(Ifis unknown use s instead.)Usenonparametric methods,which don’trequire a normaldistribution.Usenormal distribution withx - µx/nZ(This case is rare.)Use the Student t distributionwithx - µxs/ntStart
The larger Studenttcritical valueshows that with a small sample,the sample evidence must bemoreextremebefore we consider thedifference is significant.
A company manufacturing rockets claims to use anaverage of 5500 lbs of rocket fuel for the first 15 seconds ofoperation. A sample of 6 engines are fired and the mean fuelconsumption is 5690 lbs with a sample standard deviation of250 lbs. Is the claim justified at the 5% level of significance?2. Two tail t test, n < 30, unknownpopulation standard deviation3.t critical for 5% for a two tail test with 5 d.f. is 2.01586216250550056904.nsx.t5.Fail to reject HO, there is no evidence at the .05 levelthat the average fuel consumption is different from µ =5500 lbs2.015-2.0151.8621. HO:µ = 5500HA: µ5500
P-Value MethodTable A-3 includes only selected values ofSpecificP-values usually cannot be foundUse Table to identify limitsthat contain theP-valueSome calculators and computer programswill find exactP-values
P-Value Methodof Testing Hypothesesvery similar to traditional methodkey difference is the way in which we decide toreject the null hypothesisapproach finds theprobability(P-value) of gettinga result and rejects the null hypothesis if thatprobability is very low

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