Basics of Probability and Statistics Week 1 Challenge

Ans 1. The formula for the Variance of population is:`sigma^2=(sum(x-mux)^2 )/N`.

The formula for the Variance of sample is:`s^2=(sum(x-mu)^2)/(N-1)`.

It is clear through the formula that sample variance is higher than population variance as N>N-1 and hence when these factors are in the denominator, N-1 will give a higher value of variance.

If the sample variance is large then there is a chance that it captures the true(unbiased) population variance. Whereas dividing by (n) is called a biased sample estimate.

Ans 2.

In Stratifies Sampling,

• the member of strata is chosen by random selection.
• the sample is chosen by performing simple random sampling on the strata.
• the strata should be distinct and non-overlapping.
• Used for high precision(Equal Opportunity to be part of the sample)

In Cluster Sampling,

• the clusters are naturally formed from the population.
• Random selection of a cluster helps form the sample set.
• the clusters should be made distinct and non-overlapping.
• Used for cost-effective sampling.

Ans 3. The no. of different samples considering sample size to be x (without replacement) that can be created out of a population of size n disregarding the order is `= (n!)/((n-x)!* x!)`

Ans 4.

Req. Probability of drawing 2 Jacks from a pack of cards.

No. of jacks =4

No. of cards in a pack=52

Probability= Desired Outcome/ Total no. of outcomes.

Probability of drawing two cards out of deck = 52C2

No. of ways of selecting Jacks = 4C4=4 ways.

Hence probability of 2 jacks from pack of card = `p=(4*3)/(52*51)`=1/221.

Answer= 1/221

Ans 5.

Total no. of outcomes on rolling of 2 dice = 36

No. of favourable outcomes = 6 ; {(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}

Probability p = 6/36 = 1/6.

Answer = 1/6

Ans 6.

Variance is the measure of average square difference from the mean value.

Standard Deviation is the measure of the square root of variance.

Even though Variance can be used to find dispersion of data, standard deviation helps in giving the spread of data around the mean. It helps in determining the measure of uncertainty as it helps point out how far is the normal standard deviation from the expected value.