Basics of Probability and Statistics Week 1 Challenge

1.Variance formula for population = `sum(x-mu)^2 /N`

Variance formula for sample = `sum(x-mu)^2 /( N-1)`

The difference in the formula is (N-1). If we take some sample data to calculate the sample variance will give the accurate 

answer than population variance.

2. Stratified sampling : It is an improvement of systematic sampling , in this method the population data is divided into strata on the basis of some characteristics and from each of  these are further divide in smaller  homogeneous group taken randomly.

example : In a collage the population is first divided into two one is teaching and other are students . With the above method the teaching is again seperated (strata or subgroup)by teaching staff and non teaching staff .The students are divided into(strata) smaller groups like mechanical,civil,computers and electrical branches . We have to take this method by taking the sub groups  by homogeneous and within each strata heterogeneous. With the increase of this we can get the result accurately.

Clustered sampling: It is the most efficient random sampling method than the stratified method, here the data or population is first divided into clusters  and then sample is selected from the clusters randomly . The more homogeneity cluster the lesser will be the margin of errors and vice-versa.

 example : We want to the average weight of womens in state.It is not possible to take everyone's weight . So we divided the area into clusters and draw  sample randoms from these clusters.

3. Here the "n" is the size of the population and it is not the defined number to take the samples.So,if we consider "N" is the size of certain subgroup data the "N1" will be the another subgroup data size and so on . With this we can take a formula like

N1+N2+N3.....samples we can draw from size "n".

4. The total cards = 54cards

the number of jacks = 4

the probabilty of drawing 2 jacks from cards = 4C2/54 = 4*3/2*1*54 = 6/54 = 1/9

5. When two dice are drawn the number of combinations  = 36

   for getting the same number on both the dice the possibilities are = 6

   which are (1,1),(2,2),(3,3),(4,4),(5,5),(6,6)

   The probability that both the numbers on dice are same while rolling 2 dice = 6/36 = 1/6

6. The standard deviation gives the result accuracy  during the measure of despersion . The formula is

   `sum(x-mu)square div(N-1)`.For example the heights of five people are 50,40,60,60,50`

  Lets consider sample data to differentiate them `sigma ^2 = sum(x-mu)^2/N`

`mu = sumx/N`= 50+40+60+60+50/5 = 260/5 = 52`

`sigma^2 =`(50-52)^2 /5 = (-2)^2/5`=4/5=sqrt(4/5) = 2/sqrt5`

`sigma^2 = (60-52)^2/5 =(8)^2/5` = 64/5 = sqrt(64/5) = 8 /sqrt5`

standard deviation

`s^2 = (x-mu)^2/(N-1)= (50-52)^2/(5-1) = (-2)^2/4 = 4/4 = 1 =sqrt 1 = 1`

`s`s^2 = (60-52)^2/(5-1) = 8^2/4 = 64/4 = sqrt(64/4) = 8/2 = 4`

From the above results we can find that to calculate the better way to get the result accurate is with standard deviation to 

measure the despersion