Basics of Probability and Statistics Week 1 Challenge

1. There is a difference in the formula of variance for population and sample because

σ² = (Xi - mean)^2/N => Population Variance

s² = (Xi -mean)^2/(N-1) => Sample variance

The difference in denominator of the sample variance occurs because in a sample, the first n-1 values are independent of each other, but not the last value of the sample

2. In stratified sampling, the sample is randomly selected from the group or segment of data, and in each of this segment, the data is one of a kind.

Where as in clustered sampling, the data is randomly selected from a group of data which is naturally formed, called as a cluster.

3. From a population of size n, the number of possible samples depends upon the size of the sample. If the size of the sample is r, then the number of possible random samples without replacement will be C(n,r) or nCr, where as the number of random samples with replacement will be n^r.

4. The probability of drawing two jacks from a pack of cards is

4/52*3/51 = 2/26*3/51

Because, there is one jack in each suit, that is 4 suits of a pack, implies there are 4 jacks in a pack of cards. When one jack is taken out from a pack, the probabilty will be total number of jack divided by the number of cards. Then when draeing the second jack from the same pack, there will be 3 jacks and 53 total cards left, therefore the probabilty of drawing another jack will be 3/53. So the total probability will be 4/52*3/51

5. The probability that both the numbers on dice are same while rolling 2 dice is 1/6 because, the total possibilities of both dice havong same number is 6 i.e. (1,1) (2,2) (3,3) (4,4) (5,5) (6,6). And the total possible outcomes are 36, therefore required probabilty will be 6/36 = 1/6.

6. Standard deviation is required even when variance can measure the dispersion in data because in the formula of variance, to get positive values, the sum of dispersion is squared, which then alters the units of the data. So to attain the natural units of the data back, we calculate the square root of variance i.e. the standard deviation, to get the dispersion in the datav with proper units.