Basics of Probability and Statistics Week 1 Challenge

Ques1.

The difference arises due to the concept of degrees of freedom. We divide by n-1 instead of n because we are estimating the population variance from a sample. Using n-1 (called Bessel's correction) corrects the bias in the estimation, as the sample mean is already used in the formula and therefore one degree of freedom is lost.

Ques2.

Stratified Sampling:

Population is divided into strata (groups) and samples are taken from each stratum. Strata are internally homogeneous. Random sampling within each stratum. Ex- Divide by gender, take sample from each. Ensures representation from all groups.

Clustered Sampling:

Population is divided into clusters, and some clusters are randomly selected. Clusters are internally heterogeneous. Random selection of entire clusters. Ex- Select a few cities and survey all households. Economical and convenient for large populations.

Ques3.

The number of different samples depends on the sample size (r) and whether sampling is with or without replacement:

• Without replacement (Combinations): No. of samples=n! / [r! (n-r)!]
• With replacement: No. of samples=n^r

If we want all possible non-empty samples from a population of size:

Total samples = 2^n -1

Ques4.

P = 4/52 * 3/51 = 1/21 = 0.00452

Ques5.

P= 6/36 = 1/6 = 0.1667

Ques6.

While variance does measure dispersion, its unit is the square of the original unit, which is not intuitive. For example, if data is in kilograms, variance is in square kilograms.

Standard deviation is the square root of variance:

This brings the measure back to the original unit of data, making interpretation easier and more meaningful. Also, many statistical techniques use standard deviation (e.g., in normal distribution, z-scores).