Basics of Probability and Statistics Week 1 Challenge
1. Why there is a difference in the formula of variance for population and sample
The main difference between population variance and sample variance relates to the calculation of variance. Variance is calculated in five steps. First mean is calculated, then we calculate deviations from the mean, and thirdly the deviations are squared, fourthly the squared deviations are summed up and finally this sum is divided by number of items for which the variance is being calculated
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 the denominator in the formula for variance in the case of sample data is ‘n-1’, and it is ‘n’ for population data.
2. Difference between stratified and clustered sampling
Stratified sampling:
Stratified sampling is one, in which the population is divided into homogeneous segments, and then the sample is randomly taken from the segments.
Randomly selected individuals are taken from all the strata.
The selection of population elements are individual
Clustered sampling:
Cluster sampling refers to a sampling method wherein the members of the population are selected at random, from naturally occurring groups called 'cluster'.
All the individuals are taken from randomly selected clusters.
The selection of population elements are collectively
3. How many different samples can be created out of a population of size n
If considering the Null/empty set then 8 samples and if not Null/empty it is 7.
For n size it would be (n-1)!
4. What is the probability of drawing 2 jacks from a pack of card
If there are 52 cards, and 2 of them are Jacks, the probability of drawing 2 Jacks is 4c2/52c2.
Let A be the event of drawing a 1st jack from a pack of cards, so P(A) = 4/52
Case-1 With replacement.
Let P(B) be probablity of drawing a 2nd jack form a pack of card, so P(B) = 4/52
And P(A and B) = P(A) × P(B) = 4/52× 4/52 = 1/169
Case 2: without replacement
P(B) =4/51
P(A and B) = 4/52× 4/51= 4/663
5. What is the probability that both the numbers on dice are the same while rolling 2 dice
When two dice are rolled together then total outcomes are 36 and
Sample space is
[ (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) ]
So, pairs with same numbers are (1,1) (2,2) (3,3) (4,4) (5,5) (6,6) i.e. total 6 pairs
Total outcomes = 36
Favorable outcomes = 6
Probability of getting pair with same numbers = Favorable outcomes / Total outcomes = 6 / 36 = 1/6
So, P(N, N) = 1/6.
6. Why standard deviation when variance can measure the dispersion in data
The variance measures the average degree to which each point differs from the mean. It is mathematically denoted as (σ2)
While standard deviation is the square root of the variance, variance is the average of all data points within a group. It is mathematically denoted as (σ)
The standard deviation, SD is the number which gives information about the spread of data values from the mean value. If SD is small, the data values are close to the mean value. If SD is high, the data values are widely spread out from the mean value.
To bring back the unit in same we will take the square root which is known as SD