Basics of Probability and Statistics Week 1 Challenge
Q1. Answer:
The formula for sample variance is given as S²=∑(xi−μ)²/n−1
The formula for sample variance is given as σ²=∑(xi−μ)²/n
To put it simply (n−1) is a smaller number than (n). When you divide by a smaller number you get a larger number. Therefore when you divide by (n−1) the sample variance will work out to be a larger number.
Let’s think about what a larger vs. smaller sample variance means. If the sample variance is larger, it is more likely to capture the true population variance. That is why when you divide by (n−1) we call that an unbiased sample estimate. Whereas dividing by (n) is called a biased sample estimate.
Because we are trying to reveal information about a population by calculating the variance from a sample set we probably do not want to underestimate the variance. Basically, by just dividing by (n) we are underestimating the true population variance, that is why it is called a biased estimate.
Basically comes down to calculating a biased vs. unbiased sample variance estimate.
Q2. Answer:
| BASIS FOR COMPARISON | STRATIFIED SAMPLING | CLUSTER SAMPLING |
|---|---|---|
| Meaning | 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 'clusters'. |
| Sample | Randomly selected individuals are taken from all the strata. | All the individuals are taken from randomly selected clusters. |
| Selection of population elements | Individually | Collectively |
| Homogeneity | Within-group | Between groups |
| Heterogeneity | Between groups | Within-group |
| Objective | To increase precision and representation. | To reduce cost and improve efficiency. |
Q3. Answer:
A population is an entire group that you want to draw conclusions about.
A sample is a specific group that you will collect data. The size of the sample is always less than the total size of the population.
Simple random sampling (SRS) is a method of selection of a sample comprising of n a number of
sampling units out of the population having N number of sampling units such that every sampling
the unit has an equal chance of being chosen.
The samples can be drawn in two possible ways.
1. The sampling units are chosen without replacement because the units that are chosen are not
placed back in the population.
2. The sampling units are chosen with replacement because the selected units are placed back
in the population.
1. Simple random sampling without replacement (SRSWOR):
SRSWOR is a method of selection of n units out of the N units one by one such that at any stage of
selection, any one of the remaining units has the same chance of being selected, i.e. 1/ N
2. Simple random sampling with replacement (SRSWR):
SRSWR is a method of selection of n units out of the N units one by one such that at each stage of
selection, each unit has an equal chance of being selected, i.e., 1/ N
Q4. Answer:
The probability of drawing two consecutive jacks without replacement is
1 out of 221 or 1/221
Explanation:
The probability of drawing the initial jack is 4 out of 52 as there are 4 jack in deckof 52 cards
Since you are not replacing the jack the deck now has 51 cards and 3 jacks making 3 out of 51
Ist jack = 4/52
2nd jack= 3/51
To find the probability of consecutive events we would multiple the individual probabilities
4/52 x 4/51 = 12/265
which reduced to 1/221
Q5. Answer:
When two dice are rolled together then total outcomes are 36 and
Sample space is
[ (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
(4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
(5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
(6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) ]
So, pairs with same numbers are (1,1) (2,2) (3,3) (4,4) (5,5) (6,6) i.e. total 6 pairs
Total outcomes = 36
Favorable outcomes = 6
Probability of getting pair with same numbers = Favorable outcomes / Total outcomes = 6 / 36 = 1/6
So, P(N, N) = 1/6.
Q6. Answer:
Standard deviation can simply be defined as the observations that get measured are measured through dispersion within a data set.
Standard deviation is expressed in the same units of the data available.
Variance is nothing but the average taken out of the squared deviations.
Variance is expressed in Squared units.