Basics of Probability and Statistics Week 1 Challenge

1. Difference in the Formula of Variance for Population and Sample:

Population Variance ( σ^2): σ2=1/N
N∑ i=1(xi−μ)2

where, N is the population size and μ is the population mean.

Sample Variance (s^2): s2=1n−1n∑i=1(xi−¯x)2

where, n is the sample size and ¯x is the sample mean.

The adjustment in the sample variance formula (n−1 instead of N) corrects bias, ensuring a better estimation of the population variance from a sample.

2. Difference Between Stratified and Clustered Sampling:

Stratified Sampling:

• Definition: Divides the population into subgroups or strata based on certain characteristics.
• Selection: Random samples are then taken independently from each stratum.
• Advantages: Ensures representation from each subgroup, reducing variability.

Clustered Sampling:

• Definition: Divides the population into clusters, often based on geographical proximity.
• Selection: Randomly selects entire clusters, and then samples are taken from within those clusters.
• Advantages: Cost-effective, especially when clusters are naturally occurring.

3. How Many Different Samples Can Be Created out of Population of Size
n:

The number of different samples that can be created from a population of size
n is given by 2^n. Each element in the population can either be included or excluded, leading to 2 possibilities for each element.

4. Probability of Drawing 2 Jacks from a Pack of Cards:

A standard deck has 44 jacks. The probability of drawing the first jack is 452​, and the probability of drawing the second jack, given that the first was drawn, is 351​. The overall probability is the product of these probabilities: 452×351.

5. Probability That Both Numbers on Dice Are the Same While Rolling 2 Dice:

There are 6 possible outcomes on each die. The probability of getting the same number on both dice is 636=16, as there are 6 favorable outcomes out of 36 possible outcomes.

6. Why Standard Deviation When Variance Can Measure the Dispersion in Data:

While variance measures the average squared deviation from the mean, standard deviation provides a more interpretable measure by taking the square root of the variance. Standard deviation is expressed in the same units as the original data, making it more intuitive and easier to compare across different datasets.

This ensures that the measure of dispersion is on the same scale as the data, facilitating a better understanding of the spread in the dataset.