Question

The distribution of sample means has the same mean as the underlying population, but the standard deviation differs. Use a real world scenario to explain why it makes sense the variation decreases as the sample size increases.

Answer 1

Solution Suppose there are 500 students in a college. so these 500 students from the population for the research we study the avg score of the population hence the poormeter is mean score denoted by a let the mean score W) - ja a = 15 mean Now Let us take various samples of ditt sample sizes from the population. Sample 1: Let us take a sample of 36 Students and study e their mean score. The mean score of the sample do 36 skude god study their is Bermed as sample score and is denoted by x bor. Now since the ss is 36 which is > 30, we can apply the central limit theorem to the sample. accouding to the CL thecrem the sample mean is equal to the population mean and the sample or standard deviation is eaual to the The Pop's Sta. dev divided by root of 56. Does on applying CLT it to it we get to know that: Sample mean - U=72

S s . Ivo . 15 V - 2.5 u t so we can observe that the sample mean Lois same, but the sample Sta. dev is diferent Let and sample size 64 (xbar) = 72 P b. e ter Hot S - 15.voy 21.875 Interpreteation we can observe that the two Samples so drawn have somple mean, but the Sta. dev go on decreasing as the size of the Sample increases zod Sample 100 leos x bar = 72 o – 15 10o = 1.5 Now Let us take a sample of Set sixe 225 basad o r X bar = 72 c = 1517225 =). o Thus we can observe that as the sample store to incorporate more and more units of the pop! in it. the variation as presented by the sta dev goes on decreases.

is the get This pop'n closer because the sample starts representing more app and the sample estimates to the Pop'n Parameter.