Basics of Probability and Statistics Week 1 Challenge

1.Why there is a difference in the formula of variance for population and sample

Variance : `sigma``σ^2`= `∑``(``Xi`-`μx`)``^2``/``N` (population)`

Sample : `S^2`= `∑``(``Xi`-`μx`)``^2``/``N-1` (sample)`

So When We Use N insted of N-1, We have an error called Statistical bias, which means that the sample variance (the estimator) is systematically different from the true population parameter (in this case, the variance).

The only differences in the way the sample variance is calculated is that the sample mean is used, the deviations is summed up over the sample, and the sum is divided by n-1.

Population variance refers to the value of variance that is calculated from population data, and sample variance is the variance calculated from sample data. Due to this value of denominator in the formula for variance in case of sample data is 'n-1', and it is 'n' for population data

2.Difference between stratified and clustered sampling

In Cluster Sampling, the sampling is done on a population of clusters therefore, cluster/group is considered a sampling unit. In Stratified Sampling, elements within each stratum are sampled. In Cluster Sampling, only selected clusters are sampled. In Stratified Sampling, from each stratum, a random sample is selected.

Relatedly, in cluster sampling you randomly select entire groups and include all units of each group in your sample. However, in stratified sampling, you select some units of all groups and include them in your sample. In this way, both methods can ensure that your sample is representative of the target population.

3.How many different samples can be created out of population of size n

A good maximum sample size is usually around 10% of the population, as long as this does not exceed 1000. For example, in a population of 5000, 10% would be 500. In a population of 200,000, 10% would be 20,000. This exceeds 1000, so in this case the maximum would be 1000.

No. It is N raised to the power n, if you count different orderings as being different samples. If you only count the samples without order, then there are even fewer samples, but it becomes complicated to count, because the number if arrangements of a given sample depends on the pattern of repeats. Thus, each un ordered sample has n! orderings if all are distinct as is the case when drawing without replacement, but for instance the letters aabbbcccc have only 9!/2!3!4! arrangements. In any case, the answer is at most N raised to the power n, which is clearly finite.

What is infinite is the set of possible sample sizes because with replacement in a population of size N you can have a sample of any size n even much greater than N, so the set of all samples with replacement is infinite, whereas sampling without replacement you cannot have a sample of size larger than N.

4.What is the probability of drawing 2 jacks from a pack of card

The chance of drawing one of the four jacks from a standard deck of 52 cards is 4/52;

but the chance of drawing a second jacks is only 3/51,

because after we drew the first jack, there were only three jacks among the remaining 51 cards. Thus,

the chance of drawing an jacks on each of two draws is 4/52 × 3/51, or 1/221.

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

Hence, the probability of getting same number on both the dice is 1/6

The probability of two dice being the same particular number is 1/6 x 1/6 = 1/36.

This is not the same as saying that both dice are the same number. There are six different possible numbers, so that would be 6/36 or 1/6.

6.Why standard deviation when variance can measure the dispersion in data

Standard deviation (SD) is the most commonly used measure of dispersion. It is a measure of spread of data about the mean. SD is the square root of sum of squared deviation from the mean divided by the number of observations. This formula is a definitional one and for calculations, gives an actual value to how much the numbers in a data set vary from the mean. Standard deviation is the square root of the variance and is expressed in the same units as the data set.