Basics of Probability and Statistics Week 1 Challenge

Week 1 Challenge

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

• Population Variance: To calculate the population variance, you need data for the entire population, which means you have information on every individual or item of interest. In practice, it's often difficult or impossible to collect data from an entire population, especially when dealing with large populations. However, in certain scenarios where the population size is relatively small or when data on the entire population is available, the population variance can be computed directly using the formula mentioned earlier.

• Population Variance (σ²) = Σ [(Xi - μ)²] / N (or) `sigma^2 = sum [(Xi - mu)^2] / N`

  Where:

  • σ² is the population variance.
  • Σ represents the summation symbol, which means we sum up the values of (Xi - μ)² over all the data points in the population.
  • Xi represents each data point in the population.
  • μ is the population means (average).
  • N is the total number of data points in the population.

• Sample Variance: The sample variance, on the other hand, only requires data from a subset of the population, known as a sample. A sample is a proportion or subset of the entire population, and it is often more feasible and practical to collect data from a sample rather than attempting to gather data from the entire population. Researchers frequently use sampling techniques to draw representative samples from the population, and then they can calculate the sample variance using the formula mentioned earlier with the sample data.

• Sample Variance (s²) = Σ [(Xi - x̄)²] / (n - 1) or `s^2 = sum [(Xi - bar x)^2] / N-1`

  Where:

  • s² is the sample variance.
  • Σ represents the summation symbol, which means we sum up the values of (Xi - x̄)² over all the data points in the sample.
  • Xi represents each data point in the sample.
  • x̄ is the sample mean (average).
  • n is the sample size (total number of data points in the sample).

2. Difference between stratified and clustered sampling.

Stratified Sampling:

• In stratified sampling, the population is divided into non-overlapping subgroups called strata. The strata are formed based on some shared characteristic or attribute within the population. These characteristics might be age, gender, income level, geographical location, etc. The goal is to ensure that each stratum represents a specific proportion of the total population and that the variability within each stratum is minimized.
• After dividing the population into strata, a random sample is drawn independently from each stratum. The sample sizes from each stratum can be proportional to the size of the stratum (proportional allocation) or can be allocated differently, depending on the research requirements.
• Stratified sampling is useful when there is significant heterogeneity within the population, and it allows researchers to ensure that all important subgroups are adequately represented in the sample. It often leads to more precise and efficient estimates compared to simple random sampling when the strata are chosen appropriately.

Clustered Sampling:

• In clustered sampling, all the elements within the selected clusters are included in the sample. That is, all members of the selected clusters are sampled. Therefore, clustered sampling can be more efficient in terms of data collection logistics when the clusters are natural groups or are geographically close together.
• Clustered sampling is useful when the cost of collecting data on individual elements in the population is high, and it is more practical and cost-effective to gather information from clusters. However, it can lead to higher sampling variability compared to stratified or simple random sampling, especially if the clusters are highly heterogeneous.
• Clustered Sampling: Clustered sampling involves dividing the population into clusters or groups. Unlike strata, clusters are selected in a way that they are similar to the entire population, not necessarily by some shared characteristic. The goal is to create a sampling framework that simplifies data collection.

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

The number of different samples that can be created out of a population of size "n" depends on the sampling method and the size of the sample you want to draw. There are two common sampling methods:

1. Sampling with Replacement: In sampling with replacement, each time an element is selected from the population, it is put back into the population before the next selection. This means that the same element can be chosen multiple times in a single sample. In this case, the number of different samples of size "k" that can be created from a population of size "n" is given by:

                                 Number of samples with replacement = `n^k` or n^k 

Where:

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

2. Sampling without Replacement: In sampling without replacement, each time an element is selected from the population, it is not returned to the population, and hence, it cannot be chosen again in the same sample. In this case, the number of different samples of size "k" that can be created from a population of size "n" is given by the combination formula:

                               Several samples without replacement = nCk = n! / (k! * (n - k)!) 

Where:

• "n" is the population size.
• "k" is the sample size.
• "!" denotes the factorial function (e.g., 5! = 5 * 4 * 3 * 2 * 1).

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