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In order to make statistical interence (hypothesis testing and confidence interval's construction), for difference between averages of two independent populations, when both sample sizes are large (>40) we can use the Z test (with known population standard deviations, or their estimators sample standard deviations if population standard deviations are unknown). However, when sample sizes are small we have two situation : First, happens when we can assume (either based on F test, or by prior knowledge) that two population have non-equal standard deviations. In this case, we can still use a similar test statistics to the Z test statistics while the distribution is is student t distribution, and we need to calculate the degree of freedom by Welch's correction. Also, similar to the case when sample sizes are large and population standard deviations are unknown which are estimated and replaced by their estimators; we use sample standard deviations. Second case is when (we have a reason to) assume that the two population standard deviations are the same. This is called pooled procedure. 1. Pooled t procedure : a)Assume that X,p..,Xņı and Y1,..,Y»2 are samples from two independent populations X, and Y. Intuitively explain that in this case, if we were to estimate the common population standard deviation, degrees of freedom for these two samples of n;+n2 is n,+nz-2. ( Hint: There is a restriction on each of the sample deviations ; that both of deviation sets must add up to 0.). b) Open the following formula of pooled sample variance as sum of summation of deviation squared of first and second sample, that is divided by the degree of freedom in a). (n1 – 1)S7 + (n2 – 1)S n1 + n2 – 2 DISTRIBU

In order to make statistical interence (hypothesis testing and confidence interval's construction), for difference between averages of two independent populations, when both sample sizes are large (>40) we can use the Z test (with known population standard deviations, or their estimators sample standard deviations if population standard deviations are unknown). However, when sample sizes are small we have two situation : First, happens when we can assume (either based on F test, or by prior knowledge) that two population have non-equal standard deviations. In this case, we can still use a similar test statistics to the Z test statistics while the distribution is is student t distribution, and we need to calculate the degree of freedom by Welch's correction. Also, similar to the case when sample sizes are large and population standard deviations are unknown which are estimated and replaced by their estimators; we use sample standard deviations. Second case is when (we have a reason to) assume that the two population standard deviations are the same. This is called pooled procedure. 1. Pooled t procedure : a)Assume that X,p..,Xņı and Y1,..,Y»2 are samples from two independent populations X, and Y. Intuitively explain that in this case, if we were to estimate the common population standard deviation, degrees of freedom for these two samples of n;+n2 is n,+nz-2. ( Hint: There is a restriction on each of the sample deviations ; that both of deviation sets must add up to 0.). b) Open the following formula of pooled sample variance as sum of summation of deviation squared of first and second sample, that is divided by the degree of freedom in a). (n1 – 1)S7 + (n2 – 1)S n1 + n2 – 2 DISTRIBU

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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1. Pooled t procedure : In order to make statistical interence (hypothesis testing and confidence interval's construction), for difference between averages of two independent populations, when both sample sizes are large (>40) we can use the Z test (with known population standard deviations, or their estimators sample standard deviations if population standard deviations are unknown). However, when sample sizes are small we have two situation : First, happens when we can assume (either based on F test, or by prior knowledge) that two population have non-equal standard deviations. In this case, we can still use a similar test statistics to the Z test statistics while the distribution is is student t distribution, and we need to calculate the degree of freedom by Welch's correction. Also, similar to the case when sample sizes are large and population standard deviations are unknown which are estimated and replaced by their estimators; we use sample standard deviations. Second case is when (we have a reason to) assume that the two population is called pooled procedure. a)Assume that X,1,...,Xn1 and Y1..,Yn2 are samples from two independent populations X, and Y. Intuitively explain that in this case, if we were to estimate the common population standard deviation, degrees of freedom for these two samples of n,+n2 is n¡+n2-2. (Hint: There is a restriction on each of the sample deviations ; that both of deviation sets must add up to 0.) b) Open the following formula of pooled sample variance as sum of summation of deviation squared of first and second sample, that is divided by the degree of freedom in a). (n1 – 1)S + (n2 – 1)S ni + n2 – 2 Assuming that two population have the same variance this formula is like pooling two samples into one big - DISTRIBU sample and find the variance of it.
Transcribed Image Text:1. Pooled t procedure : In order to make statistical interence (hypothesis testing and confidence interval's construction), for difference between averages of two independent populations, when both sample sizes are large (>40) we can use the Z test (with known population standard deviations, or their estimators sample standard deviations if population standard deviations are unknown). However, when sample sizes are small we have two situation : First, happens when we can assume (either based on F test, or by prior knowledge) that two population have non-equal standard deviations. In this case, we can still use a similar test statistics to the Z test statistics while the distribution is is student t distribution, and we need to calculate the degree of freedom by Welch's correction. Also, similar to the case when sample sizes are large and population standard deviations are unknown which are estimated and replaced by their estimators; we use sample standard deviations. Second case is when (we have a reason to) assume that the two population is called pooled procedure. a)Assume that X,1,...,Xn1 and Y1..,Yn2 are samples from two independent populations X, and Y. Intuitively explain that in this case, if we were to estimate the common population standard deviation, degrees of freedom for these two samples of n,+n2 is n¡+n2-2. (Hint: There is a restriction on each of the sample deviations ; that both of deviation sets must add up to 0.) b) Open the following formula of pooled sample variance as sum of summation of deviation squared of first and second sample, that is divided by the degree of freedom in a). (n1 – 1)S + (n2 – 1)S ni + n2 – 2 Assuming that two population have the same variance this formula is like pooling two samples into one big - DISTRIBU sample and find the variance of it.
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