Basics of Probability and Statistics Week 1 Challenge

1) The diffrence in the formula of variance for population and sample:

Mean for population μ=∑Xi/N

Mean for Sample `x = ∑ (X i) /N`

Variance for population `σ 2 = 1 /N ∑ ( X i − μ x ) 2`

Variance for Sample `S 2 = 1 /(N − 1) ∑ ( X i − μ x ) ^2`

The main difference between population variance and sample variance relates to variance calculation. Variance is a numerical value that shows how widely the individual values in a data set distribute themselves about the mean. Population variance refers to the value of variance that is calculated from population data, and sample variance is the variance calculated from sample data. 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

2) Difference between Startified sampling and Clustered Sampling Techniques

Stratified Sampling: is a category under probability sampling which is based on dividing a population into strata, and members for the sample selected randomly from these strata.

*Elements within each stratum are sampled, from each stratum, a random sample is selected.

* Widely used to increase precision to reduce error

Clustered Sampling : In this technique population is divided into smaller groups known as Clusters. From these groups, then clusters are selected randomly to form a sample.

*It is often used to study large populations

*Used to reduce cost & increase efficiency

3) Out of population size 'n', different samples can be created using formula `n C r = (n ! )/(( n − r ) ! r !)`

4) Drawing a 2jacks from a pack of cards:

If pack has 52 cards and 4 jacks in it, probability of getting 2 jacks = E = 2

P(E)= n(E)/n(s)

n(E)=no of favourable outcomes = 52 cards - 4 jacks = 48

n(S) sample space = total no of outcomes = 52

Probability of first card being a jack = 4/52

Probability of next card also being a jack when 1 jack is already drawn = 3/51

Hence, total proabability of both the cards being a jack,
P(E=2) = 4/52 * 3/51

Probabilty of drawing 2 jacks
P(E=2) = 1/13 * 1/17 = 1/221
P(E=2) = 0.00452

5) Probability that both the numbers on dice are same while rolling 2 dice

When two dice are tossed, the possible outcomes (s), are (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)

possible outcomes n(s)=36

But we want the desired event E that the number should same on both dices.

desired outcomes n(E)=6

Probability P= n(E)/n(s) = 6/36 = 1/6 is the answer

6)

Variance is the squared value of Standard deviation. Variance (`σ 2`) gives the squared values like m^2/sec^2, while standard deviation (`σ`) gives m/s. It is a scale of measure for population

A variance is a numerical value that shows how widely the individual figures in as set of data distribute themselves about the mean.

*the vairance measures average degree to which each point differs from the mean - the average of all data points while standard deviation spread out of a group numbers is from the mean

*The calculation of variance uses square because it weighs ouliers more heavily than data closer to mean. This calculation also prevents differences above the mean from canceeling out those below, which would result in a variance of zero.