Basics of Probability and Statistics Week 1 Challenge

1) Why there is a difference in the formula of variance for population and sample

Ans:-

The sample variance is an estimate of  σ² and is very useful in situations where calculating the population variance would be too cumbersome. The only difference in the way sample variance is calculated is that the sample mean is used, the deviations are summed up over the sample, and the sum I divided by n-1. When calculating sample variance, n  is the number of sample points (vs N for population size in the formula above).

Unlike the population variance, the sample variance is simply a statistic of the sample. it depends on the research methodology and on the sample chosen. A new sample or a new experiment will likely give you a different sample variance, although if your samples are both representative your sample variance should be good estimates of the population variance and so close to each other.

The population and sample variance is show above the there is a small change in the denominator, the population variation has denominator n which is the number of item in the population and for sample variance is one less i.e n-1.The calculation sample variance is slightly higher than the calculated population varience.

population variance σ² = ∑(x1-µx)^2 /N

sample variance S² = ∑(x1-µx)^2 /(N-1)

2) Difference between stratified and clustered sampling

points                                                 Stratified                                                       Cluster sampling

Defination               It is a sampling technique in which the population                It is a sampling Technique in which the 

                             is divided into subgroups or strata. Here the sample,            population is not divided manually into any

                            are then extracted randomly from every group created.         groups.Here the samples are randomly 

                                                                                                                     picked from the naturally formed group

                                                                                                                    termed as clusters.

purpose                To increase precision and representation                               To reduce cost and improve efficiency.

Divergency            Done by the researcher or group of researchers                   The clusters occur naturally forming subgroup

Sample selection    The sample is taken up randomly from all the manually       The sample is taken up randomly from all over

                            created subgroups or strata.                                               the naturally formed population clusters.

Heterageneity        The sample are taken from between the manually created    The samples are taken within the naturally,                                   strata.                                                                               developed group or cluster.

Homogeneity          The samples are taken from within the artificially                 The sample in cluster sampling are taken 

                             created subgroup.                                                              from different natural clusters no 

                                                                                                                     diversification in population.

uses                      Diversification in population                                                 No diversification in population

Pece                     Stratified sampling is slower                                                 Cluster sampling is relatively faster.

3) How many different samples can be created out of population of size n

The size of the sample to be taken is equal to the size of the population. This means that the number of different random samples that can be taken is 1.

A good maximum sample size is usually around 10% of the population, as long as does not exceed 1000.For example ,in a population of 5000, 10% would be 500. In a population of 200000, 10 % would be 20000, this exceeds 1000 so in this case the maximum would be 1000.Even in a population of 200000 sampling 1000 people will normally give a fairly accurate result.

Sampling more than 1000 people won`t add much to the accuracy given the extra time and money it would cost.

 4) What is the probability of drawing 2 jacks from a pack of card

      1 card can be drawn in 52C1 ways

                     n= 52C1

                       =52                 

drawing a two jack

4C2 =(4*3/2*1) = 6

p= 6/52

p= 3/26

1)Getting two cards out of 52 is 

              52C2=(52x51/2x1)=1326

2) Getting 2 jacks out of 4 jacks is

              4C2=(4x3/2x1)=6

probability of getting 2 jacks out of 52 cards is given by

P=favorable outcomes/total outcomes

            P =6/1326

             P=1/221

5) What is the probability that both the numbers on dice are same while rolling 2 dice

  The number of outcomes while rolling 1 dice = 6

   Number of outcomes while rolling 2 dice = 6^2

                                                              = 36

The sample space for rolling a dice is given 

{ (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 ni=umber 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

6) Why standard deviation when variance can measure the dispersion in data

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 number in a data set varies from the mean. standard deviation is the square root of the variance and is expressed in the same units as the data set.

standard deviation is the most commonly used measure of dispersion. it is a measure of the spread of data about the mean. standard deviation is the square root of the sum of square deviation from the mean divided by the number of observations.                                

population variance σ² = ∑(x1-µx)^2 /N

standard deviation  σ = ∑(x1-µx)^2 /N

The Variance  is the average of squared  deviations and standard deviation is the square root of Variance.

1)So Standard deviation is best measure of dispersion.

2)Variance is the best measure for statistical analysis.