Basics of Probability and Statistics Week 1 Challenge

• Difference between stratified and clustered sampling

Aspect	Stratified Sampling	Cluster Sampling
Sampling Strategy	Population divided into distinct subgroups	Population divided into naturally occurring clusters
Objective	Reduce variability and increase representativeness	Increase efficiency and ease of data collection
Selection Process	Random sampling within each stratum	Random sampling of entire clusters
Variability	Lower variability within strata, higher between strata	Higher variability within clusters, lower between clusters
Precision	Increased precision for subgroup analysis	Decreased precision for subgroup analysis
Sample Size	Larger sample size required	Smaller sample size compared to individual sampling
Data Analysis	Stratified analysis required	Cluster-based analysis required
Complexity	More complex and time-consuming	Less complex and relatively easier
Cost	Potentially higher cost due to larger sample size	Potentially lower cost due to smaller sample size
Representativeness	Ensures representation of each stratum	May not fully represent all individual elements
Examples	Surveying students from different grade levels	Surveying households in specific neighborhoods

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

To use the population variance you need all of the data available whereas to use the sample variance you only need a proportion of it. For example, if we take ten words at random from this page to calculate the variance of their length, a sample variance would be needed.

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

The number of different samples of size n that can be selected from a finite population of size N is given by the combination formula, which is denoted as NCn. This formula is defined as NCn=N! n!

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

The probability that the first will be a Jack is 4/52, or 1/13.

Without replacement, the probability that the second card will be a Jack given that the first is a Jack is 3/51, or 1/17.

Multiplying these two values

the probability that both draws will be Jacks. 1/13 x 1/17 = 1/221 ans.

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

Sample points of getting same number on both dice

(1,1) ,(2,2) ,(3,3) ,(4,4) ,(5,5) & (6,6).

Hence, the probability of getting same number on both the dice is 1/6.

• Why standard deviation when variance can measure the dispersion in data

standard deviation is preferred over variance because standard deviation can be compared with the mean. The concept of variance has a central role in statistics because it is used to calculate other statistics like ANOVA, R-Squared, hypothesis testing, statistical inference, and more.