Basics of Probability and Statistics Week 1 Challenge

Ans 1 :- Formula

population variance :- `sigma^2` = `sum (x i - mu x)^2/N`

Sample Variance :- `s^2 = sum(x i -mu x)^2 /(N-1)`

Population variance refers to the value of variance that is calculated from population data, and sample variance is the variance that is calculated from sample data. it means it has few data from whole data suppose their is a forest which has 1000 tree and you have to anaylize this so it is very time cosuming and expensive to go through with 1000 tree so we took few tree like 50 and then anlaysis it Due to this value of denominator in the formula for variance in case of sample data is ‘n-1’, and it is ‘n’ for population data. As a result both variance and standard deviation derived from sample data are more than those found out from population data

Ans 2 :-

1. In Stratified Sampling Population is Separated into different homogeneous segments called strata, and then the sample is chosen from the each stratum randomly, is called Stratified Sampling. Cluster Sampling is a sampling technique in which the units of the population are randomly selected from already existing groups called cluster.
2. In stratified sampling the individuals are randomly selected from all the strata, to constitute the sample. On the other hand cluster sampling, the sample is formed when all the individuals are taken from randomly selected clusters
3. In cluster sampling, population elements are selected in aggregates, however, in the case of stratified sampling the population elements are selected individually from each stratum.
4. In stratified sampling, there is homogeneity within the group, whereas in the case of cluster sampling the homogeneity is found between groups.
5. Heterogeneity occurs between groups in stratified sampling. On the contrary, the members of the group are heterogeneous in cluster sampling.
6. When the sampling method adopted by the researcher is stratified, then the categories are imposed by him. In contrast, the categories are already existing groups in cluster sampling.
7. Stratified sampling aims at improving precision and representation. Unlike cluster sampling whose objective is to improve cost effectiveness and operational efficiency

Ans 3 :- n! samples can be created out of population of size n

Formula:- n! = n* n-1 * n-2 *- - - * 2 * 1

Ans 4:- probability of drawing 2 jacks from a pack of card

Their are three case for this

first card is jack * second is not jack + first is not jack * second is jack + first is jack * second is jack

So According to this probality is 1/52 * 51/52 + 51/52 * 1/52 + 1/52 * 1/52

=> 51/2704 + 51/2704 + 1/2704

=> 103/2704

so probability of drawing 2 jacks from a pack of card is 103/2704

Ans 5:- probability for both the numbers on dice are same while rolling 2 dice is

total case is (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)

(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)

so favourable events = (1,1),(2,2),(3,3),(4,4),(5,5),(6,6)

so probability = 6/36 => 1/6

Ans 6:- We can't take negetive value as result so negelat that value we use standard deviation