Basics of Probability and Statistics Week 1 Challenge
Basics of Probability and Statistics Week 1 Challenge
1. Why there is a difference in the formula of variance for population and sample.
Population variance: ∑(xi-u)^2/n
Sample variance: ∑(xi-u)^2/(n-1)
In sample varience we are dividing by n-1 instead of n, we are using a slightly smaller denominator, which increases the value of the variance estimate,making it more accurate as an estimator of the population variance. When we use n in the denominator, the resulting variance estimate will be an underestimate of the population variance, which means that it will be biased.
2. Difference between stratified and clustered sampling.
| Stratified sampling | Clustered sampling. |
| Stratified sampling is one, in which the population is divided into homogeneous segments, and then the sample is randomly taken from the segments | Cluster sampling refers to a sampling method wherein the members of the population are selected at random, from naturally occurring groups called 'cluster'. |
| Stratified sampling is slower | Clustered sampling is relatively faster |
| The purpose of the Stratified sampling is to increase the precision | The purpose of the Clustered sampling is to increase the precision |
| Homogeneity is within the group | Homogeneity is between groups |
| Heterogeneity is between groups | Heterogeneity is within the group |
| Example: For population of the state, Male ,Female and other strata | Example: If you were to conduct a study on the consumption of soda in a particular city, you could use area sampling to divide the city into different areas, called clusters, and then select certain clusters to be a part of the sample group. |
3. How many different samples can be created out of population of size n.
i) If sampling is done without replacement the number of possible samples is given by the combination formula:
n! / ((n-r)! * r!)
Where n is the population size and r is the sample size. This formula calculates the number of ways to choose r items from a set of n items, without regard to order.
ii) If sampling is done with replacement , the number of possible samples is given by the formula:
n^r
Where n is the population size and r is the sample size. This formula calculates the number of ways to choose r items from a set of n items, allowing for duplicates.
4.What is the probability of drawing 2 jacks from a pack of card.
i) Probability of drawing 2 jacks from a pack of card with replacement.
4/52 x 4/52 = 0.0059
There are still 4 jacks out of 52 cards, but the probability of drawing a jack on the second draw is still 4/52, because the first jack is replaced back into the deck before the second draw. So the probability of drawing 2 jacks is simply the product of the probability of drawing a jack on the first draw and the probability of drawing a jack on the second draw.
ii) Probability of drawing 2 jacks from a pack of card with replacement.
4/52 x 3/51 = 0.0045
There are 4 jacks out of 52 cards. The probability of drawing the first jack is therefore 4/52. After the first jack is drawn, there are only 51 cards left in the deck, and 3 of those are jacks. So the probability of drawing the second jack, given that the first jack has already been drawn, is 3/51. Multiplying these probabilities together gives the overall probability of drawing 2 jacks.
5.What is the probability that both the numbers on dice are same while rolling 2 dice
There are 6 possible outcomes for each die, so there are 6 × 6 = 36 possible outcomes for rolling two dice. Out of these, there are 6 outcomes where both dice show the same number: (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6).
6/36 = 1/6 = 0.1667
6.Why standard deviation when variance can measure the dispersion in data.
Variance is the average of the squared differences from the mean, while standard deviation is the square root of the variance. In other words, variance is calculated by averaging the squared deviations of each data point from the mean, while standard deviation is calculated by taking the square root of the variance to return the measurement back to the original unit of measurement.