Week1B&2B2304 - 1 ADM2304: One Sample Mean Dr....

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Unformatted text preview: 1 ADM2304: One Sample Mean Dr. Suren Phansalker 1. Sample Mean: Nomenclature: Population Sample Parameters Statistics Size: N n Mean: Variance: 2 s 2 StdDev: s Y 2 If you are curious and want to know the Theory about how to create Confidence Intervals and Test Hypotheses about Population Means, please read Lecture2AMTB2304. Thanks to the Central Limit Theorem (CLT), we know that the sampling distribution of sample means is Normal with mean and standard deviation . Remember that when working with Population Means, the mean and standard deviation are NOT linked. This implies that knowing tells us nothing at all about . The best thing for us to do is to estimate with s , the sample standard deviation. This gives us . ( ) SD Y n = Y ( ) SD Y ) ( SE Y s n = 3 William S. Gosset, an employee of the Guinness Brewery in Dublin, Ireland, worked long and hard to find out what the sampling model was. The sampling model that Gosset found was the t-distribution, known also as Students t . T h e t-distribution is an entire family of distributions, indexed by a parameter called degrees of freedom. We often denote degrees of freedom as df , and the model as t df . The t-distribution is a relatively complex mathematical relation involving Integration and most people find it convenient to tabulate it rather than actually do the Integration! 4 2. Once Again CLT for : If , the Population Standard Deviation is known, then: , a Normal Distribution. Generally, , is unknown and must be estimated by s. Then: Since , one can rewrite the above as: , a t-Distribution with (n-1) degrees of freedom. Y , Y N n 1 , n s Y t n ( ) s SE Y n = 1 [ , ( ) ] n Y t S E Y 5 3. Distribution for Sample Means: Written differently; When the conditions are met, the standardized sample mean follows a Students t-model with n 1 degrees of freedom. We estimate the standard error with: 1 ( ) n Y Y t S E = ( ) s SE Y n = 6...
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Week1B&2B2304 - 1 ADM2304: One Sample Mean Dr....

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