Basics of Probability and Statistics Week 1 Challenge
Ans 1. Population variance: Uses N because you're working with the full dataset, no bias.
σ2=∑(xi−μ)^2/N
Sample variance: Uses n−1 to adjust for the bias that arises from using a sample to estimate
population variance.
s2=∑(xi−μ)^2/n-1
Ans 2.
| Characteristic | Stratified Sampling | Cluster Sampling |
|---|---|---|
| Population Division | Divides population into homogeneous groups (strata) | Divides population into heterogeneous clusters |
| Sampling Method | Randomly samples from each stratum | Randomly selects entire clusters |
| Within-Group Variation | Low (similar members within strata) | High (clusters resemble population diversity) |
| Between-Group Variation | High (strata are different) | Low (clusters are similar) |
| Objective | To ensure representation of all subgroups | To simplify sampling and reduce costs |
Ans 3. For Without Replacement. Number of possible samples: n! / (n-k)! (where k is the sample size)
For With Replacement Number of possible samples: n^k (where k is the sample size)
Ans 4. 1/221
Ans 5. 1/6
Ans 6. While variance measures dispersion, standard deviation is generally preferred due to its interpretability, comparability, stability, and mathematical convenience. However, variance has its own uses in specific contexts.
->Standard deviation is more stable and less sensitive to outliers than variance.
-> Standard deviation allows for direct comparison across different datasets or variables with different units. Variance does not.