Basics of Probability and Statistics Week 1 Challenge

Q.1:-

When we use N instead of N-1, we have an error called statistical bias, which means that the sample variance is systematically different from the true population parameter.

If we repeat this process many times by taking a different (but same sized) sample each time and averaged the results the result for the mean would indeed be close to the population mean. However, the sample variance would be underestimated (biased). If we multiply the variances by n/(n−1) then the sample variances would also match the population variance.

Q.2:-

In cluster sampling we just randomly select cluster (group) and in stratified sampling we randomly select members from strata.

In stratified sampling all groups are included, But in cluster sampling we randomly choose the group

In stratified sampling homegenity within group, Heterogeneity between groups And in cluster sampling homegenity between groups, Heterogeneity within group

in cluster sampling we randomly select entire groups and include all units of each group in your sample. However, in stratified sampling, you select some units of all groups and include them in your sample. In this way, both methods can ensure that your sample is representative of the target population.

Q.3:-

samples from population of n size =2^n

If we are not considering null set then= 2^n-1

Q.4:-

the chance of drawing the first card X The chance of drawing the second card.

since the first card can be any of the 4 jacks, the chances of drawing one of them is= 4/ 52

the second card can be any of the remaining jacks, though there are now 51 cards in the deck so the chances of the second card also being a jack are =3/51

The chances of getting two jacks are

(4*3)/(52*51) = 12/2652 = 1/221

So the chances of drawing the same rank in two consecutive cards is 1/221

Q.5:-

Since, the number of outcomes while rolling a dice = 6

Number of outcomes while rolling two dice = 6*6 = 36

The Sample Space for rolling a dice is given as,

{(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)}

Sample points of getting same number on both dice :-

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

Thus, the number of favourable outcomes = 6

Total number of outcomes = 36

P (Getting same number on both dice) = 6/36

= 1/6

Hence, the probability of getting same number on both the dice is 1/6.

Q.6:-

Variance helps to find the distribution of data in a population from a mean, and standard deviation also helps to know the distribution of data in population, but standard deviation gives more clarity about the deviation of data from a mean.

In the definition of standard deviation and Variance we can say Standard deviation measures how far apart numbers are in a data set. Variance, on the other hand, gives an actual value to how much the numbers in a data set vary from the mean. Standard deviation is the square root of the variance and is expressed in the same units as the data set.