Basics of Probability and Statistics Week 1 Challenge

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

Ans .

• Total number of cards in a pack = 52
• Totla number of jacks in a pack = 4
• probability of drawing 1 jack = 4/52
• probabiltiy of drawing 2nd jack from same pack

without replacing the card or placing it back in pack of cards is 3/52

= (4/52)*(3/52)

= 1/221

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

Ans.

• Total number of outcomes while rolling 2 dice = 36
• total number of samples having same number on both the dice = 6
• eg : (1,1),(2,2),(3,3),(4,4),(5,5),(6,6)
• probability of having both the number same when rolling 2 dice = 6/36 = 1/6

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

Ans. if we have population of size n

and if we consider sample size population of n-1

then number of different samples cn be reated is (n-1)!

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

Ans. we know the formula for variance is sigma = Σ(Xi− μ)**2/N

and for standard deviation is = √(sigma**2)

so variance gives the actual value to how much number in a data set vary from the mean.

where as std deviation tells how far appart numbers are in a data set and gives value in original unit.

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

Ans

Imagine we have a huge bookshelf. we measure the total thickness of the first 6 books and it turns out to be 158mm. This means that the mean thickness of a book based on first 6 samples is 26.3mm.
Now we take out and measure the first book's thickness (one degree of freedom) and find that it is 22mm. This means that the remaining 5 books must have a total thickness of 136mm
Now we measure the second book (second degree of freedom) and find it to be 28mm. So we know that the remaining 4 books should have a total thickness of 108mm .
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In this way, by the time we measure the thickness of the 5th book individually (5th degree of freedom) , we automatically know the thickness of the remaining 1 book.

This means that we automatically know the thickness of 6th book even though we have measured only 5. From this concept, In a sample of size n, we know the value of the n'th observation even though we have only taken (n-1) measurements. i.e, the opportunity to vary has been taken away for the n'th observation.

This means that if we have measured (n-1) objects then the nth object has no freedom to vary. Therefore, degree of freedom is only (n-1) and not n.

so the purpose of this difference is to get better and unbiased estimate of the population.

Q. Difference between stratified and clustered sampling

Ans

• Stratified sampling is performed by dividing the population into groups and then selecting some random members from the groups .
  • eg : if we have numbers from range 1-50 and we created the groups in the form of range 1-10,11-20...,41-50.
  • then we select some number from each group like 4,7 from group of range 1-10 and 32,28 from group range 31-40 and soo.
• Clustered samping is perfomed by dividing the population into equla clusters then selecting all members from some random clusters
  • eg: we have numbers from range 1-50 and we created the clusters in the from 1-10,11-20,21-30,..,41-50
  • here we will select all number from some cluster like we select 11-20, 41-50 these two clusters and all its numbers will be considered in the samples