Basics of Probability and Statistics Week 1 Challenge
1. Why there is a difference in the formula of variance for population and sample?
- The difference in the formula of variance for population and sample is due to the fact that the population variance is a parameter that describes the true variability of the entire population, while the sample variance is a statistic that estimates the population variance based on a subset of the population.
- To make the sample variance an unbiased estimator of the population variance, we divide by n-1 instead of n, where n is the sample size.
- This is known as Bessel’s correction, and it compensates for the fact that we are using the sample mean instead of the population mean to calculate the variance.
2. difference between stratified and cluster sampling.
- Stratified and clustered sampling are two types of probability sampling methods that divide the population into groups, but they have different purposes and procedures.
- Stratified sampling aims to increase the precision and representation of the sample by dividing the population into homogeneous groups (strata) based on some relevant characteristics, such as age, gender, income, etc., and then randomly selecting some members from each group to be in the sample.
- Cluster sampling aims to reduce the cost and improve the efficiency of the sample by dividing the population into heterogeneous groups (clusters) that are geographically or administratively convenient, such as schools, districts, regions, etc., and then randomly selecting some clusters and including all members from those clusters in the sample.
3. How many different samples can be created out of a population of size n?
- The answer to this question depends on whether the samples are drawn with or without replacement, and whether the order of the elements in the samples matters or not.
- If the samples are drawn with replacement, it means that each element of the population can be selected more than once in the same sample.
- If the samples are drawn without replacement, it means that each element of the population can be selected only once in the same sample.
- If the order of the elements in the samples matters, it means that two samples with the same elements but in different order are considered different.
- If the order of the elements in the samples does not matter, it means that two samples with the same elements but in different order are considered the same.
The number of different samples of size k that can be created out of a population of size n can be calculated using the following formulas:
- With replacement and order matters: n^k
- With replacement and order does not matter: (n+k-1)Ck
- Without replacement and order matters: nPk
- Without replacement and order does not matter: nCk
Where nPk and nCk are the permutations and combinations of n things taken k at a time, respectively.
4. What is the probability of drawing 2 jacks from a pack of card.
- N: the total number of cards in the pack, which is 52.
- K: the total number of jacks in the pack, which is 4.
- n: the number of cards drawn, which is 2.
- k: the number of jacks drawn, which is 2.
The formula for the hypergeometric distribution is:
P(X
=K)=(Kk)(N−Kn−k)(Nn)P(X=k)=(nN)(kK)(n−kN−K)
where ( a
b)(ba) is the binomial coefficient, which means the number of ways to choose b items from a set of a items.
Plugging in the values, we get:
P(X
=2)=(42)(52−42−2)(522)P(X=2)=(252)(24)(2−252−4)
P(X
=2)=6×11326P(X=2)=13266×1
P(X
=2)=1221P(X=2)=2211
Therefore, the probability of drawing 2 jacks from a pack of 52 cards is 12212211, which is approximately 0.0045 or 0.45% .
5. what is the probability that both the numbers on the dice are the same while rolling 2 dice?
- n: the number of dice, which is 2.
- s: the number of individual die faces, which is 6.
- N: the total number of possible outcomes when rolling n dice, which is s^n, or 6^2, or 36.
- k: the number of favorable outcomes when both dice show the same number, which is s, or 6.
The formula for the basic counting principle is:
P(E
)=kNP(E)=Nk
where P(E) is the probability of the event E.
Plugging in the values, we get:
P(both dice are same)=636P(both dice are same)=366
P(both dice are same)=16P(both dice are same)=61
Therefore, the probability that both the dice are the same while rolling 2 dice is 1661, which is approximately 0.1667 or 16.67%.
6. why standard deviation when variance can measure the dispersion in data?
Standard deviation and variance are both measures of dispersion, which means they indicate how much the data values are spread out from the mean. However, there are some reasons why standard deviation is preferred over variance in most cases:
- Standard deviation has the same unit as the original data, while variance has the squared unit. This makes standard deviation more intuitive and easier to interpret than variance. For example, if the data values are in meters, then the standard deviation is also in meters, but the variance is in square meters, which is harder to understand.
- Standard deviation is more compatible with the normal distribution, which is a common statistical model for many phenomena. The normal distribution has a property that about 68% of the data values are within one standard deviation of the mean, and about 95% are within two standard deviations of the mean. This makes standard deviation a useful measure of how likely a data value is to be close to the mean. Variance does not have such a simple relationship with the normal distribution.
- Standard deviation is more robust to outliers than variance. Outliers are extreme data values that are far away from the mean and can distort the measure of dispersion. Since variance involves squaring the deviations from the mean, it gives more weight to outliers than standard deviation, which can inflate the measure of dispersion. Standard deviation, on the other hand, is less sensitive to outliers and gives a more accurate measure of dispersion.