Basics of Probability and Statistics Week 1 Challenge

1. Why there is a difference in the formula of variance for population and sample.
Ans :
Population Variance:
The population variance measures the dispersion of a variable in the entire population. The formula for population variance is:

σ² = Σ((x - μ)²) / N

where:

σ² represents the population variance.
Σ represents the sum of.
x represents each individual value in the population.
μ represents the population mean.
N represents the size of the population.
In population variance, the deviations from the population mean are squared and averaged over the entire population size.

Sample Variance:
The sample variance estimates the dispersion of a variable based on a subset or sample of the population. The formula for sample variance is:

s² = Σ((x - x̄)²) / (n - 1)

where:

s² represents the sample variance.
Σ represents the sum of.
x represents each individual value in the sample.
x̄ represents the sample mean.
n represents the sample size.

2. Difference between stratified and clustered sampling .
Ans :
Stractified : In stratified sampling, the population is divided into non-overlapping subgroups or strata based on certain characteristics or variables. The strata are internally homogeneous but distinct from one another. The main steps in stratified sampling are:

Clustered sampling : In cluster sampling, the population is divided into clusters or groups, and a subset of clusters is selected for analysis. The clusters should be internally heterogeneous but externally similar. The main steps in cluster sampling are:

3. How many different samples can be created out of population of size n .
Ans : The number of different samples that can be created from a population of size "n" depends on the sampling method and the desired sample size.

4. What is the probability of drawing 2 jacks from a pack of card.
Ans :
To calculate the probability of drawing two Jacks from a standard deck of 52 cards, we need to consider the total number of possible outcomes and the number of favorable outcomes.

Total number of possible outcomes:
There are 52 cards in a deck, so the total number of possible outcomes is 52.

Number of favorable outcomes:
Since there are four Jacks in a deck (one Jack of Hearts, one Jack of Diamonds, one Jack of Clubs, and one Jack of Spades), the number of favorable outcomes is 4.

Probability of drawing two Jacks:
To calculate the probability, we divide the number of favorable outcomes by the total number of possible outcomes:

Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
= 4 / 52
= 1 / 13
≈ 0.0769

5. What is the probability that both the numbers on dice are same while rolling 2 dice.
Ans :

To calculate the probability of getting the same number on both dice when rolling two standard six-sided dice, we need to determine the total number of possible outcomes and the number of favorable outcomes.

Total number of possible outcomes:
Each die has six possible outcomes (numbers 1 to 6), so the total number of possible outcomes for rolling two dice is 6 x 6 = 36.

Number of favorable outcomes:
There are six possible outcomes where both dice show the same number: (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), and (6, 6). Therefore, the number of favorable outcomes is 6.

Probability of getting the same number on both dice:
To calculate the probability, we divide the number of favorable outcomes by the total number of possible outcomes:

Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
= 6 / 36
= 1 / 6
≈ 0.1667

6 Why standard deviation when variance can measure the dispersion in data .
Ans :
While variance measures the dispersion or spread of data, standard deviation is a closely related measure that is often preferred in practice for several reasons:

Standard deviation is in the same unit as the original data: Variance is calculated by squaring the differences from the mean, which results in the unit of the squared value. In contrast, the standard deviation is the square root of the variance, which brings it back to the original unit of the data. This makes the standard deviation more interpretable and relatable to the original data.