Basics of Probability and Statistics Week 1 Challenge
1) Why there is a difference in the formula of variance for population and sample
Ans:-Two Quantities that we are considering here .
PopulationVariance−σ²
SampleVariance−s²
Sample Variance of “n” samples from the Population is given by the below formula
The reason behind the Denominator being (n-1) , is that the total degrees of freedom of this sample system with n samples is n-1 .
Why??
Because , The first (n-1) samples are independent of each other given the mean, but the last one is not , you can find the last value , given the n-1 values and the mean.
2) Difference between stratified and clustered sampling
Ans:- The differences between stratified and cluster sampling can be drawn clearly on the following grounds:
1) A probability sampling procedure in which the 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.
3) How many different samples can be created out of population of size n
Ans :-It is N raised to the power n, if you count different orderings as being different samples. If you only count the samples without order, then there are even fewer samples, but it becomes complicated to count, because the number if arrangements of a given sample depends on the pattern of repeats. Thus, each un ordered sample has n! orderings if all are distinct as is the case when drawing without replacement, but for instance the letters aabbbcccc have only 9!/2!3!4! arrangements. In any case, the answer is at most N raised to the power n, which is clearly finite.
4) What is the probability of drawing 2 jacks from a pack of card
Ans :- If there are 2 Jacks in a pile of 52 cards already, and you add two more Jacks, then your probablilty of drawing out a Jack is 4 out of 54 (or 4/54), and when you divide the numerator and denominator by 2/2 (because it's a factor of 1, so you don't change the equation), you get 2/26.
5) What is the probability that both the numbers on dice are same while rolling 2 dice
Ans:- When two dice are rolled together then total outcomes are 36 and
Sample space 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, pairs with same numbers are (1,1) (2,2) (3,3) (4,4) (5,5) (6,6) i.e. total 6 pairs
Total outcomes = 36
Favorable outcomes = 6
Probability of getting pair with same numbers = Favorable outcomes / Total outcomes = 6 / 36 = 1/6
So, P(N, N) = 1/6.
6) Why standard deviation when variance can measure the dispersion in data
Ans:- The reason why standard deviation is so popular as a measure of dispersion is its relation with the normal distribution which describes many natural phenomena and whose mathematical properties are interesting in the case of large data sets.