Basics of Probability and Statistics Week 1 Challenge
Answer no.1
The formula for population variance is:
σ² = Σ(x-μ)²/N
where σ² is the population variance, Σ(x-μ)² is the sum of squared deviations of each data point from the population mean μ, and N is the total number of data points in the population.
On the other hand, the formula for sample variance is:
s² = Σ(x-x̄)²/(n-1)
where s² is the sample variance, Σ(x-x̄)² is the sum of squared deviations of each data point from the sample mean x̄, and n is the total number of data points in the sample.
The main difference between these two formulas is the denominator: for population variance, we use N, the total number of data points in the population, while for sample variance, we use n-1, which is the number of degrees of freedom in the sample.
The use of n-1 in the sample variance formula is due to the fact that when we estimate the variance of a sample, we are estimating it from a smaller set of data than the entire population. This means that the sample variance tends to underestimate the true population variance. By using n-1 instead of n as the denominator, we can correct for this bias and obtain an unbiased estimate of the population variance.
Answer no.2
Stratified sampling involves dividing a population into non-overlapping subgroups, or strata, based on some relevant characteristic, such as age, gender, income, or geographic location. The strata are selected so that they are internally homogenous, meaning that the individuals within each stratum are similar to each other, but are different from individuals in other strata. Then, a simple random sample is selected from each stratum, with the size of each sample proportional to the size of the corresponding stratum. This ensures that the sample is representative of the entire population with respect to the stratifying variable.
Clustered sampling, on the other hand, involves dividing the population into non-overlapping groups, or clusters, based on some natural grouping, such as households, schools, or neighborhoods. Then, a simple random sample of clusters is selected, and all individuals within the selected clusters are included in the sample. This method is often used when it is more feasible or cost-effective to sample groups rather than individuals.
Answer no.3
The number of different samples that can be created out of a population of size n depends on the size of the sample and whether or not sampling is done with replacement.
If sampling is done without replacement, meaning that each element can be selected only once, then the number of possible samples of size k that can be selected from a population of size n is given by the binomial coefficient:
n! / (k!(n-k)!)
where "!" denotes the factorial function. This formula gives the number of ways to choose k objects from a set of n distinct objects, without regard to their order.
For example, if we have a population of size n=10, and we want to select a sample of size k=5 without replacement, then the number of possible samples is:
10! / (5!(10-5)!) = 252
This means that there are 252 different ways to select a sample of 5 elements from a population of 10 elements, without selecting the same element twice.
Answer no.4
There are 4 jacks in the deck, so the probability of drawing a jack on the first draw is 4/52.
On the second draw, there will be 51 cards left in the deck, but only 3 jacks remaining. So the probability of drawing another jack is 3/51.
To find the probability of both events occurring together (drawing two jacks in a row), you multiply the probabilities of each individual event.
So the overall probability of drawing two jacks from a deck of cards is:
(4/52) * (3/51) = 0.0045 or approximately 0.45%
Therefore, the probability of drawing two jacks from a standard deck of 52 cards is about 0.45%.
Answer no.5
When rolling two fair six-sided dice, the probability that both dice will show the same number is calculated as follows:
There are 6 possible outcomes for the first die, and for each of these outcomes, there is one and only one matching outcome on the second die, for a total of 6 possible outcomes for the second die. Therefore, there are 6 * 1 = 6 possible outcomes where both dice show the same number.
There are 36 possible outcomes in total when rolling two dice, since each die has 6 possible outcomes. So, the probability of rolling two dice and getting the same number on both dice is:
6 / 36 = 1/6
Thus, the probability of rolling two dice and getting the same number on both dice is 1/6 or approximately 0.1667.
Answer no.6
Variance is a measure of how spread out the data is, and it is calculated by taking the average of the squared differences between each data point and the mean of the dataset. The variance is expressed in squared units, which can make it difficult to interpret.
Standard deviation, on the other hand, is the square root of the variance and is expressed in the same units as the original data. By taking the square root of the variance, we obtain a measure of dispersion that is easier to interpret and communicate.