Basics of Probability and Statistics Week 1 Challenge
1. Why there is a difference in the formula of variance for population and sample
sample variance : variance of a sample from the population. Calculated by dividing the sum of the squared deviations from the mean by the sample size minus 1.
population variance : variance of the entire population. Calculated by dividing the sum of the squared deviations from the mean by the population size
2. Difference between stratified and clustered sampling
Stratified sampling divides the population into homogeneous groups, or strata, based on a shared characteristic such as age, gender, or income. A random sample is then selected from each stratum. This ensures that the sample is representative of the population as a whole, even if the population is heterogeneous.
Cluster sampling divides the population into clusters, which are groups of individuals that are naturally associated with each other. For example, a cluster might be a school, a neighborhood, or a company. A random sample of clusters is then selected, and all members of the selected clusters are included in the sample. This method is often used when it is difficult or expensive to contact individuals directly.
3.How many different samples can be created out of population of size n
- Sampling with replacement means that each member of the population can be selected more than once. In this case, the number of different samples that can be created is n^n. For example, if the population size is 3, then there are 27 different samples that can be created with replacement: {1, 1, 1}, {1, 1, 2}, {1, 1, 3}, ..., {3, 3, 3}.
- Sampling without replacement means that each member of the population can only be selected once. In this case, the number of different samples that can be created is nCr, where nCr is the combination of n taken r. For example, if the population size is 3, then there are 6 different samples that can be created without replacement: {1, 2, 3}, {1, 3, 2}, {2, 1, 3}, {2, 3, 1}, {3, 1, 2}, {3, 2, 1}.
| Population size | Sampling with replacement | Sampling without replacement |
|---|---|---|
| 2 | 4 | 1 |
| 3 | 27 | 6 |
| 4 | 81 | 20 |
| 5 | 243 | 70 |
| 6 | 729 | 210 |
4.What is the probability of drawing 2 jacks from a pack of card
- There are 4 jacks in a deck of 52 cards, so the probability of drawing one jack is 4/52 = 1/13.
- Since the cards are not replaced, the probability of drawing a second jack after drawing a first jack is 3/51.
- So, the probability of drawing 2 jacks in a row is (1/13) * (3/51) = 0.004524886877828055.
5.What is the probability that both the numbers on dice are same while rolling 2 dice
- There are 6 possible outcomes for each die, so there are a total of 6*6=36 possible outcomes for 2 dice.
- There are 6 ways to roll the same number on 2 dice, so the probability of rolling the same number is 6/36=1/6.
- Since the dice are independent, we can multiply the probability of rolling the same number on each die to get the probability of rolling the same number on both dice, which is (1/6)*(1/6)=1/36.
6.Why standard deviation when variance can measure the dispersion in data
Variance and standard deviation are both measures of dispersion, which means they measure how spread out the data
- Variance is the square of the standard deviation. This means that the variance is in squared units, which can make it difficult to interpret.
- Standard deviation is in the same units as the data itself. This makes it easier to interpret and compare standard deviations from different datasets.
- Standard deviation is more sensitive to outliers than variance. This means that standard deviation will be more affected by extreme values in the data.