Basics of Probability and Statistics Week 1 Challenge

1. The key difference between population and sample variance formulae lies in the denominator.

Population Variance

Denominator: N ( the total number of individuals in the population)

formulae = σ² = Σ(xᵢ - μ)² / N

Sample Variance

Denominator: n-1 (the sample size minus 1)

Formulae = s² = Σ(xᵢ - x̄)² / (n - 1)

Reason for the difference

When Calculating the sample variance, we are using the sample mean (x̄) as an estimate of the population mean (μ). This estimate is inherently less precise the true population mean. To account for this uncertainty and to provide a more accurate estimate of the population variance, we use n-1 in the denominator.

2. Feature Stratified Sampling Clustered Sampling

Strata Homogenous Heterogenous

Sampling From each Stratum Within selected Clusters

Goal Representation of subgroups Reduced cost and effort

3. The number of different samples that can be created out of a population of size n depends on the sample size, k

If the sampling is done without replacement:

This mean that once an individual is selected, they cannot be selected again for the same sample.

The number of possible samples is given by the combination formula:

C(n, k) = n! / (k! * (n-k)!)

If the sampling is done with replacement:

This means that an individual can be selected multiple times for the same sample.

The number of possible samples is given by the permutation formula:

P(n, k) = n^k

Therefore, the number of different samples depends on whether the sampling is done with or without replacement and the desired sample size.

4. There are 52 cards in a standard deck of card, and 4 of them are jacks.

There are 4 jacks out of 52 cards, so the probability of drawing a jack on the first draw is 4/52.

Probability of drawing the second jack:

After drawing one jack, there are 3 jacks left out of 51 cards. So , the probability of drawing a jack on the second draw is 3/51.

Probability of drawing 2 jacks in a row:

To find the probability of both events happening we multiply the probabilities.

4/52 * 3/51 = 12/2652 = 1/221

5. Probability of rolling the same number on two dice

There are 6 slides on a die, and each side has a unique number.

Total possible outcomes : 6*6 = 36

So, the probability of rolling the same number on two dice is:

6/36 = 1/6

6. Standard deviation is preferred over variance because it's easier to interpret

Variance, being a squared value, doesn't have the same units as the original data. Standard deviation, as the square root of variance, returns a value in the same units, making it more comparable to the mean and other data points. This makes it more intuitive and easier to understand the spread of the data in relation to its average.