Basics of Probability and Statistics Week 1 Challenge

Question: Why is there a difference in the formula of variance for population and sample?

First of all, we need to understand what we call the population, sample, and variance. So first let's start with the population.

What is the population?

" Population is the entire set of data from which we will be using some of the data for our statistical analysis.

What is the sample?

"sample is the data we have selected from the population according to various sampling techniques. Samples are also used when the data for the population is very large.

The formula for the Population variance is `sigma^2 = [underset(i=1)overset(n)(Sigma) (x i-mu)^2]/N`

Where the formula for the sample variance is `sigma^2 = [underset(i=1)overset(n)(Sigma) (x i-mu)^2]/(N-1)`

It is divided by the 'N-1' instead of N due to vessel correction which will give you the correct value of the variance of the population which we are interested in anyway.

You should calculate the sample variance if you are working with data taken from a larger population.

You should calculate the population variance if you are working with the data set of the entire population.

Question: Difference between stratified and cluster sampling

Sampling is the technique to select the data such that it will represent the targeted population. The quality of the sampling data can lead to great results if the sampling represents the population closely. Stratified and cluster sampling is the technique to select the sample data from the population on the basis of probability.

Stratified Sampling: In this sampling, the population is divided into strata and the sample data are selected from this stratra.In order to divide the population data to stratra there are several rules which strata must satisfied the rules are the strata must be homogenous and also collectively exhaustive and mutually exclusive as well.The strata must define a part of population.

Also the members of sample must be distinct that means the member is only a part of one strata, that means that require entire population must be part of the sample.

Cluster sampling: Cluster sampling is used when the homogeneity is external but the heterogeneity is internal within clusters or groupings.it is process of dividing the population in multiple groups.It reduces the number of interviews and cost to reach desired accuracy.when the major heterogeneity is within the cluster the chance of random error is less.

Question : how many different samples can be created from the population of size n?

if the population size is N and the size of samples taken from the population is n then ,we can get `(NCn ) = (N!)/[((N-n)!) * n!]`

Question : what is the probability of drawing a 2 jacks from the pack of cards ?

Total cards in the packs = 52

Total jackes in the packs is = 4

probability of drawing first jack = `(((4),(1)))/(((52),(1)) ] = 4/52 = 0.076`

now if it is without replacement = `(((3),(1)))/(((51),(1)) ] = 3/51 = 0.058`

so the probability of getting two jacks in a raw = 0.076*0.058 = 0.004408 = 0.4 %

and if two cards are drawn at a same time than

probability is

`(((4),(2)))/(((52),(2)) ] = 6/1326= 0.00405` = 0.4 %

Question : probability of both dice have same number if 2 dies are rolled

each dice has 6 number

so number of outcomes from rolling the 2 dice = 6^2 = 36

sample space = (1,2) (1,3) (1,4) (1,5) (1,6) (1,1)

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

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

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

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

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

The desire output numbers are = 6

so the probability is = 1/6

Question :Why standard deviation when variance can measure the dispersion in data

in variance, we have squared the value to get the variation over the mean value so to maintain consistency with the units we developed the standard variation so we can compare the values of different quantities.