Basics of Probability and Statistics Week 1 Challenge
1. Why there is difference in the formula of variance for population and sample?
Answer for Question 1:
The Formula for the Population and Sample Variance is as follows,
Population Variance:`sigma^2` =(`summation`(Xi-`mux`)^ 2)/N
Sample Variance:`S^2` =(`summation`(Xi-`mux`) ^2)/(N-1)
The difference in the formula is the difference in denominator. this difference is due to the sample variance exhibits exactly one bit of information from the sample set, in the mean time the population mean does not. thus the sample mean gives one less degree of freedom to the sample set.
that's why the sample variance is called as the unbiased estimator. This we can see through the degree's freedom example.
Population Variance: suppose we draw three independent observations from a population where mean=11. so the three observation values may be of any values. after the two values the third values might be of any, so the population mean exhibits three degress of freedom.
| i | xi |
xi-mu |
| 1 | 10 | 10-11=-1 |
| 2 | 5 | 5-11=-6 |
| 3 |
but for the Sample Variance: if the mean is unknown and we find the sample mean bar X =5, then the table will be
| i | xi |
xi-mu |
| 1 | 9 | 9-5=4 |
| 2 | 4 | 4-5=-1 |
| 3 | 2 | 2-5=-3 |
from the above table it is evident that the third observation value is no more having any degree of freedom. since that it has one less degree of freedom than the population variance.
Thus concludes the difference in the formula for the population variance and sample variance.
2. Difference between stratified and clustered sampling?
Answer for Question 2:
The difference between the startified and clustered sampling is mentioned as follows:
Stratified Sampling:
1. Stratified sampling means in which the population is divided based on similar characters(lets say homogenous) and then the sample is randomly taken from the segments which was already created.
2. Randomly selected individuals are taken from all the Strata. Strata means the groups or segments which was divided based on the homogenous properties. so in the sampling there may be more number of strata, from that individuals are selected randomly.
3. The Selection of population elements is taken place in individual manner.
4. The Bifurcation or grouping was imposed by the researcher.
5. The important use of the Stratified sampling is to increase the precision and representation.
6. The Stratified sampling has two types:
a) Proportionate Stratified sampling
b) Dis-Proportionate Stratified sampling
Clustered Sampling:
1. Clustered sampling means in which the population are selected at the random, from naturally occuring groups called clusters.
2. All the individuals are taken from the randomly selected clusters. so in the sampling there may be more number of clusters, from that clusters one cluster is selected and individuals are selcted randomly from that cluster.
3. The Selection of population elements is taken place in Collective manner.
4. The Bifurcation or segrication was by the naturally occuring groups.
5. The important use of the Stratified sampling is to reduce the cost and improve efficiency.
6. The Clustered sampling has Three types:
a) Single Stage Clustered sampling
b) Two Stage Clustered sampling
1) Simple Cluster.
2) Probability Proportionate to size sampling.
c) Multi-Stage Clustered sampling
Thus concludes the difference between the Stratified and Clustered sampling.
3. How many different samples can be created out of population of size n?
Answer for Question 3:
From the population Size n we can create 2^n samples.
if we ignore original sample then we can create 2^(n-1) samples.
if at all we ignore the original and null sample then we can create 2^(n-2) samples.
4. What is the probability of drawing 2 jacks from a pack of cards?
Answer for Question 4:
The Probability of getting of 2 Jacks from the pack of cards is as follows,
the Pack contains 52 cards, so we need to get 2 jacks from the 52 cards. by using the probability formula `52P2`= 52! / (1*2)
= 52 * 51 / (2)
= 2652 / 2
= 1326.
so the probability of drawing 2 jacks for the pack of cards is 1326.
5. What is the probability that both the numbers on dice are same while rolling 2 dice?
Answer for Question 5:
The dice has six faces. we are rolling two dice so the total possible outcome will 6^2=36.
the possible outcomes is mentioned as follows,
| Dice | 1 | 2 | 3 | 4 | 5 | 6 |
| 1 | (1,1) | (1,2) | (1,3) | (1,4) | (1,5) | (1,6) |
| 2 | (2,1) | (2,2) | (2,3) | (2,4) | (2,5) | (2,6) |
| 3 | (3,1) | (3,2) | (3,3) | (3,4) | (3,5) | (3,6) |
| 4 | (4,1) | (4,2) | (4,3) | (4,4) | (4,5) | (4,6) |
| 5 | (5,1) | (5,2) | (5,3) | (5,4) | (5,5) | (5,6) |
| 6 | (6,1) | (6,2) | (6,3) | (6,4) | (6,5) | (6,6) |
so from the above table it is evident that the same numbrs are formed when rolling two dices is in the diagonal line.
i.e 6/36 = 1/6.
6. Why Standard Deviation when variance can measure the dispersion in data?
Answer for Question 6:
The Variance unit will be the square of the input units. suppose we are taking the samples as the speed of the different vehicle as 30km/h, 65Km/h and 98Km/h. so the variance units will be in the Km^2/h^2. with this units we cannot do the analysis. for the analysis the units must be consistent with the sample inputs.
to bring back the units to do the analysis, the standard deviation is created.
the formula for the S.D `sigma`= `sqrt sigma^2`