Basics of Probability and Statistics Week 1 Challenge

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

A. The formula for Variance for Population is

`sigma`^2 = `Sigma`(`X i`- `mu`)^2 /N

The formula for Variance for sample is

`sigma`^2 = `Sigma`(`X i`- `mu`)^2 /(N-1)

The difference between these formulas is a small change in the denominator. i.e., The denominator of sample Variance is reduced by 1.

The calculated value of the Sample Variance is high when compared with the variance for population. Because due to the lack of information available regarding the population, we increase the variance of sample data. i.e., (By decreasing the denominator, numerator increases).

2. Difference between stratified and clustered sampling

A. Stratified Sampling: This method involves forming stratas or groups from a population based on a distinct characteristics .For determing the statistics, we take equal numbers of entities from each stratas. For examples, if there are 3 different colors of baskets in the population, we separate the baskets based on the color. Later a definite number of baskets are taken for each color and statistics are calculated. In stratified sampling, each strata is equally represented in the sample generated.

Clustered Sampling: This method isused when mutually homogeneous yet internally heterogeneous groupings are evident in a statistical population. In this sampling plan, the total population is divided into these groups (known as clusters) and a radaom sample of the groups is selected. The elements in each cluster are then sampled. If all elements in each sampled cluster are sampled, then this is referred to as a "one-stage" cluster sampling plan. If a simple random subsample of elements is selected within each of these groups, this is referred to as a "two-stage" cluster sampling plan. A common motivation for cluster sampling is to reduce the total number of interviews and costs given the desired accuracy.

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

A. For Example, let us consider the size of population is 3

`X`= {1,2,3}

Now, the possible outcomes are {1}{2}{3}{1,2}{1,3}{2,3}{1,2,3}

Here the total samples are : 7; We compare with `2^n`-1.

We are subracting with 1, because we are not taking the null vector.

So, Total number of different samples can be created out of population of size n is `2^n-1`

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

A. We know that, the pack of card contains 52 cards.

Now we have to draw 2 cards out of 52.

`52c2`= `(52*51)/2`= 1326

We also know that A pack of cards contains 4 jacks,

`4c2`= `(4*3)/2`= 6

Now, Proabilty = No of Favourable outcomes / Total outcomes

Probabilty = 6/1326

= 1/221

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

A. We know that, when two dice are rolled , The total outcomes are 36

Now, the probability that both the numbers on dice are same is, The possible outcomes are

{1,1}{2,2}{3,3}{4,4}{5,5}{6,6}

Fovurable outcomes : 6

Now, Proabilty = No of Favourable outcomes / Total outcomes

Probabilty = 6 / 36

= 1 / 6

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

A. Variance is calculated by suaring the difference of variable from its mean. The problem here is dimensions are different for mean and Variance.

So To overcome the problem, Standard Deviation is introduced, It is square of the Variance. By squaring of the variance the mean and standard deviation are dimensionally equal.