Basics of Probability and Statistics Week 1 Challenge

Assignment -1

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

• There is a difference in the formula in the denominator. (n-1) is used in the denominator to reduce the bias. There are two types of sample variances: Biased & Unbiased variances, we use n-1 in the denominator to reduce the variation of the sample variance from the true mean (Population mean). As shown in figure 1.1 when we use n instead on n-1, we are underestimating the variance. (It is below the true variance). When we use n-1. The sample variance is close to the true variance. When we use n+1 we over-estimate the variance of the sample. As a sample represents a population it is essential that the sample variance is closed to the true variance of the population, So using n-1 gives a variance that is closer to the population variance.

Fig:1.1

2. Difference between stratified and clustered sampling?

Stratified sampling: In Stratified sampling, A sample is drawn from each strata using a random sampling method like simple random sampling or systematic sampling.

Fig 2.1

The samples are bifurcated into various mutually exclusive, homogenous, and non-overlapping subgroups. In stratified sampling, samples are picked from different categories of data (which have a common attribute). It is more unbiased and representative. In figure 2.1, The three categories (strata) are represented as blue, green, and red. The samples from each category have been selected using simple random sampling.

Clustered Sampling: Clustered sampling is defined as a sampling technique in which the population is divided into already existing groups called clusters.

Fig 2.2

Then a sample of the cluster is selected randomly from the population. The term cluster refers to a natural, but heterogenous, intact grouping of the members of the population. Commonly used variables in the clustering population are geographical areas, buildings, schools, etc. Heterogeneity of the cluster is an important feature of an ideal cluster sample design.

3. How many different samples can be created out of a population of size n?

If the population has n elements in it, Then the number of samples that can be created is 2ⁿ

Where n is the number of individual elements in the population. For example:

If A {1, 3, 5} is the population containing 3 elements then

The subset of A containing no elements - { } (Null set)

The subset of A containing one element each - {1} {3} {5}

The subset of A containing two elements each - {1, 3} {1, 5} {3, 5}

The subset of A containing three elements - {1, 3, 5)

Therefore, all possible subsets of A are { }, {1}, {3}, {5}, {1, 3}, {3, 5}, {5,1}, {1, 3, 5}

Therefore, number of all possible subsets of A is 8 which is equal 2³ including the null set.

Proper subsets are = {3}, {5}, {1, 3}, {3, 5}, {5,1}, {1, 3, 5} (without the null set)

Number of proper subsets are 7 = 8 - 1 = 2³ – 1

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

4C₂ is the number of favorable outcomes as there are 4 jacks in 52 cards and the probability of selecting two jacks out of 4 is 4C₂.

The probability of selecting 2 cards out of 52 cards is 52C₂.

So, Probability = Number of Favorable outcomes/Number of possible outcomes

= 4C₂/52C₂

= (4! /2! *2!)/ (52! /2! *50!)

= 6/1326 = 1/221

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

Total number of favorable outcomes (numerator) = 6 ({1,1},{2,2},{3,3},{4,4},{5,5},{6,6})

Total number of possible outcomes (denominator) = 6² = 36

Probability = 6/36 = 1/6

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

Variance is an intermediate stage to obtain standard deviation. Variance can measure the dispersion of data, but the units are in the second power which do not match the original units of data.

For example, if we are calculating the standard deviation of speed and variance of speed for a set of data. The units of variance will km²/ h² which do not match with the original data units whereas the standard deviation will have the same units of the original data, in this instance, it will be km/h which is the same unit system of original data.