Basics of Probability and Statistics Week 1 Challenge

1.Why there is a difference in the formula of variance for population and sample

The Formula for Variance of Sample is

`s^2 = (sum_(i=1)^n (x_i-mu)^2)/(n-1)`

The Formula for Population Variance is

`sigma^2 = (sum_(i=1)^n (x_i-mu)^2)/N`

This used for the Concepts of degrees of Freedom.

For the population Variance we find the Variance for Total Population. So N is Used. Total Population size is N.

The sample Variance is We find the Variance for one sample in Total Sample Size.

We know all values of Sample except one we find out that one sample part we assumed like that and we use this formula n-1.

This correction made by Bessel’s for Sample Variation.

Here µ is the Mean of sample in sample variance and mean of population in population variance.

2.Difference between stratified and clustered sampling.

In Stratified Sampling Each Stratum/ or layer Elements are used for sampling.

In Clustered sampling some of clusters are used.

In Stratified Sampling Elements in each stratum or layer selected randomly for sampling.

In Clustered Sampling Clusters or groups selected randomly for sampling.

In Cluster Sampling, the aim is to reduce cost and increase the efficiency of sampling.

In Stratified Sampling, the motive is to increase precision to reduce error.

In Stratified Sampling, Homogeneity Within the group because the population is divided into homogeneous segments.

In Clustered Sampling, Homogeneity Between groups because members of the population are selected at random, from naturally occurring groups called 'cluster'.

In Stratified Sampling, elements are collected Individually.

In Clustered Sampling, elements are collected collectively.

4.What is the probability of drawing 2 jacks from a pack of Cards.

Probability = number of Favourable outcomes/ Total number of Possible outcomes

Total 52 Cards. In that 4 are jacks.

Probability of getting 1 jack is 4/52 = 1/13.

Then 3 jacks are present in that cards out of 51 cards.

Another jack probability is 3/51

Then total probability is 1/13 X 3/51

= 3/663 =1/221.

5.What is the probability that both the numbers on dice are same while rolling 2 dices.

Total number of outcomes while two dice thrown is (one dice)² = 6² = 36

Total number of outcomes in one dice = 6.

Number of possible events is = 6

The possible outcomes are

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

Then Probability of getting same dice is

= 6/36 = 1/6.

6.Why standard deviation when variance can measure the dispersion in data.

Variance will be used between variables to find how much differ to each other. And Standard deviation will use to find the Dataset or variables are how much far away from the average or mean of dataset.

And the Standard deviation gives much more clarity about deviation of data from mean.

Standard deviation =`s = sqrt((sum_(i=1)^n (x_i-mu)^2)/(n-1))`

To get the Variation into same units of mean. to get better understanding.

3.How many different samples can be created out of population of size n.

If Population size n then we create n-1 sample size possibility

If we know samples then Population Size X ⁿC_r X Sample Size.

For Example we take Population Size N =4

Then We Take Sample Size(n) Upto N-1 = 4-1 =3.

Total samples = N X ⁿC_r X samplesize

4 X ⁿC_r X3 = Total Samples.

or

Total Samples = ^{population size}Samples_{sample size} = Population Size !/((Sample size !)X(Population Size- Sample size)!