Basics of Probability and Statistics Week 1 Challenge
1 Ans:
Degree of freedom is defined as the maximum number of logically independent values, which are values that have the freedom to vary, in the data sample.
For example, We have a table containing data of 6 people. We know their mean height and height of 5 people. We can calculate the height of 6th people by using mean height and height of other 5 people. Here, the constraint is 1. So, the degree of freedom is n-1 (i.e.,) 6-1 = 5.
So, when we calculate the sample variance, we use sample mean and (n-1) instead of n. We want sample variance to be an unbiased estimator of population variance.
Mathematical fact:
The deviation around the sample mean tend to be a bit smaller than population mean. So, If we use n instead of n-1, we would be underestimating the population variance.
So, we use (n-1) instead of n, while calculating the sample variance to get an unbiased estimate of population variance
2 Ans:
In stratified sampling, the population is divided into homogeneous groups called strata, using an attribute of the samples. Then members from each stratum are selected, and the number of samples taken from those strata is proportional to the presence of the strata within the population.
In cluster sampling, the population is grouped into clusters, predominantly based on location, and then a cluster is selected at random.
In cluster sampling, a cluster is selected at random, whereas in stratified sampling members are selected at random.
In stratified sampling, each group used (strata) include homogenous members while, in cluster sampling, a cluster is heterogeneous.
Stratified sampling is slower while cluster sampling is relatively faster.
Stratified samples have less error due to factoring in the presence of each group within the population and adapting the methods to obtain a better estimation.
Cluster sampling has inherent higher percentage of e.
3 Ans:
The number of possible samples of size n=3 n = 3 that can be drawn from this population without replacement is binom {N} {n}=binom {12} {3}=220 (nN) = (312) = 220 The number of samples that can be drawn from this population with replacement is N^n=12^3=1728 N n = 123 = 1728
4 Ans:
since there are a total of 4 jacks in a deck of 52, the probability of the getting the first jack is 4:52 = 4/52 = 1/13
.
Now, after the first jack is removed, there are only 3 jacks left in a deck of 51 so the probability of the second card drawn is:
3:51 = 3/51 = 1/17
.
Therefore the probability is:
1/13 * 1/17 = 0.0045.
5 Ans:
the reduced sample space is of size = 6+(6/2)=6+15=21.
Though the sample space is reduced from 36 to 21, the probability of getting the same number on both dice is 1/36, and the probability of getting different number on both the dice is 2/36.
Since we have 6 possibilities of getting same number on both the dice, the required probability is 6/36=1/6.
6Ans:
Standard deviation is used for the statistical test to know about the relationship exist between two sets of variable Variance measures the distribution of data in a population around the central value. Standard deviation measures the distribution of data relative to the central value.