Basics of Probability and Statistics Week 1 Challenge

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

-- The formula for variance differs for population and sample data because the calculation of sample variance involves an

additional step to account for the fact that the sample is only a subset of the population, and thus may not fully represent

the characteristics of the population.

The variance is a measure of how much the data points in a distribution are spread out from the mean. The population

variance is calculated as the average squared deviation of each data point from the population mean, and is given by the formula:

σ^2 = Σ (xi - μ)^2 / N

where:

• σ^2 is the population variance
• Σ (xi - μ)^2 is the sum of squared deviations from the population mean
• N is the population size

In contrast, the sample variance is calculated as the average squared deviation of each data point from the sample mean, and is

given by the formula:

s^2 = Σ (xi - x̄)^2 / (n - 1)

where:

• s^2 is the sample variance
• Σ (xi - x̄)^2 is the sum of squared deviations from the sample mean
• n is the sample size

The key difference between the two formulas is that the denominator in the sample variance formula is (n - 1) instead of N.

This is because the use of N in the population variance formula assumes that we have data for the entire population, while

the use of (n - 1) in the sample variance formula accounts for the fact that the sample is only a subset of the population

and is likely to have some degree of sampling error. This correction is necessary to ensure that the sample variance is an

unbiased estimate of the population variance.

2.Difference between stratified and clustered sampling

-- Stratified sampling and clustered sampling are both methods of selecting samples from a larger population, but they

differ in their approach to grouping the population and selecting samples.

Stratified sampling involves dividing the population into distinct subgroups, or strata, based on certain criteria

(such as age, gender, income level, geographic location, etc.). A sample is then randomly selected from each stratum,

in proportion to the size of the stratum relative to the overall population. Stratified sampling ensures that each subgroup

of the population is adequately represented in the sample, which can help to reduce sampling error and increase the accuracy of estimates.

Clustered sampling, on the other hand, involves dividing the population into clusters or groups based on certain criteria

(such as neighborhoods, schools, businesses, etc.). A sample of clusters is then randomly selected, and all individuals

within each selected cluster are included in the sample. Clustered sampling is often more convenient and less expensive

than other sampling methods, as it can reduce the cost and time required to locate and sample individuals scattered across

a large geographic area.

The main difference between the two methods is that stratified sampling focuses on dividing the population into homogeneous

subgroups, while clustered sampling focuses on dividing the population into geographically or organizationally defined clusters.

Stratified sampling ensures that each subgroup is adequately represented in the sample, while clustered sampling can be more

efficient for sampling populations that are geographically dispersed or organized into groups or communities.

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

-- The number of different samples that can be created out of a population of size n depends on the sample size, or the number

of individuals included in each sample.

If we are selecting samples without replacement, meaning that once an individual has been selected for a sample, they cannot be

selected again, then the number of different samples that can be created is given by the formula:

n! / (n - k)! k!

where:

• n is the population size
• k is the sample size

This formula represents the number of possible combinations of k individuals from a population of n individuals.

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

-- To find the probability of drawing 2 jacks from a pack of cards, we can use the formula for the probability of a combination:

P(k items out of n) = nCk * p^k * (1-p)^(n-k)

where:

• P is the probability
• nCk is the number of combinations of k items that can be selected from a set of n items
• p is the probability of success (drawing a jack)
• (1-p) is the probability of failure (not drawing a jack)
• k is the number of items to be selected
• n is the total number of items

In a standard deck of 52 cards, there are 4 jacks, so the probability of drawing a jack on the first draw is 4/52, or 1/13. After

the first jack is drawn, there are only 3 jacks remaining in the deck, and the total number of cards has decreased to 51.

Therefore, the probability of drawing a second jack from the remaining deck is 3/51, or 1/17.

Using the formula for the probability of a combination, the probability of drawing 2 jacks from a deck of cards can be calculated as follows:

P(2 jacks) = 4C2 * (1/13)^2 * (12/13)^2

P(2 jacks) = 6 * 1/169 * 144/169

P(2 jacks) = 0.0045 or approximately 0.45%

Therefore, the probability of drawing 2 jacks from a pack of cards is approximately 0.45%.

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

-- The probability that both numbers on two dice are the same is the probability of getting any of the possible pairs of same numbers.

There are six possible outcomes for each die, and therefore there are 6 x 6 = 36 possible outcomes when rolling two dice.

Out of these 36 possible outcomes, there are 6 outcomes where both dice show the same number (i.e., (1,1), (2,2), (3,3), (4,4), (5,5), or (6,6)).

Therefore, the probability of rolling two dice and getting the same number on both is:

P(same number on both dice) = number of desired outcomes / total number of possible outcomes P(same number on both dice) = 6 / 36

Simplifying the fraction, we get:

P(same number on both dice) = 1 / 6

Therefore, the probability of rolling two dice and getting the same number on both is 1/6, or approximately 0.1667 (or 16.67%).

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

-- Although variance is a useful measure of dispersion in data, standard deviation has some advantages over variance:

1. Standard deviation is expressed in the same units as the data, making it easier to interpret. Variance, on the other hand,

3. is expressed in squared units, which can be difficult to understand and compare.

4. Standard deviation is more commonly used and widely recognized than variance, so it may be more useful for communication and comparison purposes.

5. Standard deviation is more mathematically tractable than variance, which can simplify calculations in certain situations.

6. Standard deviation is less sensitive to extreme values than variance. Since variance is based on the squared differences between each value and the mean, it is more heavily influenced by extreme values than standard deviation.

7. In some statistical tests and methods, standard deviation is required as an input or assumption, so it may be more useful to calculate and report standard deviation rather than variance.

Overall, both standard deviation and variance are useful measures of dispersion in data, and the choice of which to use may depend on the specific needs of the analysis or application.