Weekly Notes Intro Stat Week 9
Weekly Notes Intro Stat Week 9 TMATH 110 C
University of Washington Tacoma
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This 2 page Class Notes was uploaded by Qihua Wu on Sunday November 29, 2015. The Class Notes belongs to TMATH 110 C at University of Washington Tacoma taught by KENNEDY,MAUREEN C. in Fall 2015. Since its upload, it has received 21 views. For similar materials see Intro Stat Applications in Math at University of Washington Tacoma.
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Date Created: 11/29/15
Recall that in standard normal distribution we assume that we know the variance of the population yet it is uncommon in real life where we would know the variance of the population but not the mean Therefore we assume the variance of the population by using the sample variance However this then introduces an additional source of variability the variance on top of the sampling error of the mean in this case we use the tdistribution Comparing to standard normal distribution tdistribution has higher tails at the end making the occurrence of extreme values more usual because to preserve the con dence level when we have an additional source of variability we need to get larger values on the tails The tstatistic is distributed as a t distribution with sample size 1 degrees of freedom where the degrees of freedom in calculating statistics means the number of freely varying observations The reason why the degrees of freedom is sample size1 is because as we calculate the mean one value becomes xed To calculate the tstatistics for the estimated population mean based on the sample of size n instead of using variance of the population like the standard normal distribution we use the variance of the sample and instead of using 2 values we use t M tn1sampe mean population mean sample standard deviation n5 Assumptions for the tdistribution The sample mean is normally distributed either the sample itself is normally distributed or the sample size is greater than 30 The data are sampled from random sampling Properties of the tdistribution Symmetric at 0 since mean is normally distributed Goes from negative in nity to positive in nity More extreme values are seen as usual longer tails than normal standard distribution Distribution based on sample size different sample size results in different degrees of freedom which looking at the ttable result in different values which then cause different shape As degrees of freedom goes to in nity the mean approaches 0 and the variance approaches 1 because of the central limit theorem though the t distribution is still a bit more likely to have longer tails than standard normal distribution As sample size increases the critical values get smaller for a given con dence level because of central limit theorem As con dence level increase the critical values get bigger for a given sample size
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