Basics of Probability and Statistics Week 1 Challenge

• 1.Why there is a difference in the formula of variance for population and sample
• Variance: Variance is a numerical value that shows how widely the individual figures in a set of data distribute themselves about the mean. That is how far each number is from the mean, and thus from each other. A variance of zero value means all the data are identical. More the variance, more are the values spread out about mean, hence from each other. Less the variance, less are the values spread out about mean, hence from each other, and variance can’t be negative.

difference in the formula of variance for population and sample :

• The main difference between population variance and sample variance relates to calculation of variance. Variance is calculated in five steps. First mean is calculated, then we calculate deviations from the mean, and thirdly the deviations are squared, fourthly the squared deviations are summed up and finally this sum is divided by number of items for which the variance is being calculated. Thus variance= Σ(xi-x-)/n. Where xi = ith. Number, x- = mean and n = number of items..

Now, when the variance is to be calculated from population data, n is equal to the number of items. Thus if variance in blood pressure of all the 1000 people is to be calculated from data on blood pressures of all the 1000 people, then n = 1000. However when the variance is calculated from sample data 1 is to be deducted from n before dividing the sum of the squared deviations. Thus in the above

example if sample data :

can have 100 items, the denominator would be 100 – 1 = 99.

2.Difference between stratified and clustered sampling

Cluster sampling is a type of sampling method in which we split a population into clusters, then randomly select some of the clusters and include all members from those clusters in the sample.

For example, suppose a company that gives whale-watching tours wants to survey its customers. Out of ten tours they give one day, they randomly select four tours and ask every customer about their experience

Stratified sampling is a type of probability sampling, in which first of all the population is bifurcated into various mutually exclusive, homogeneous subgroups (strata), after that, a subject is selected randomly from each group (stratum), which are then combined to form a single sample. A stratum is nothing but a homogeneous subset of the population, and when all the stratum are taken together, it is known as strata.

The common factors in which the population is separated are age, gender, income, race, religion, etc. An important point to remember is that strata should be collectively exhaustive so that no individual is left out and also non-overlapping because overlapping stratum may result in the increase in the selection chances of some population elements. The sub-types of stratified sampling are:

• Proportionate Stratified Sampling
• Disproportionate Stratified Sampling

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

the population size is N=2 ,N=12.

the number of possible sample of size n=3, n=3

that can be drawn form this population without replacement is binom {N} {n}=binom{12}{3}=220(nN)=220

the number of sample that can be drawn from this population with replacememnt is N^n=12^3=1728N n=123=1728

examples

{1,2,3,4,5}

{1}{2}{3}{4}{5}

{1,2}{2,3}{3,4}{4,5}{5,1}{2,1}{2,4}{5,2}{3,5}

(1,2,3}{4,5,1}{2,3,4}{1,2,5}

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

thr probability of drawing 2 or 3 jacks from a standard deck of cards with out replacememnt p(3 jacks or 2 jacks)=p(3jacks)+p(2jacks)=p(j)

p(j/j(j/j)+p(j)p(j/j)*(3!/2!)

Number of favourable outcomes i.e. ‘a king of red colour’ is 2 out of 52 cards.

Therefore, probability of getting ‘a king of red colou

. A card is drawn from a well shuffled pack of 52 cards. Find the probability of:

(i) ‘2’ of spades(ii) a jack(iii) a king of red colour(iv) a card of diamond(v) a king or a queen(vi) a non-face card(vii) a black face card(viii) a black card(ix) a non-ace(x) non-face card of black colour(xi) neither a spade nor a jack(xii) neither a heart nor a red king

In a playing card there are 52 cards.

Therefore the total number of possible outcomes = 52

(i) ‘2’ of spades:

Number of favourable outcomes i.e. ‘2’ of spades is 1 out of 52 cards.

Therefore, probability of getting ‘2’ of spade

Number of favorable outcomes
^{P(A) =} Total number of possible outcome

= 1/52

(ii) a jack

Number of favourable outcomes i.e. ‘a jack’ is 4 out of 52 cards.

Therefore, probability of getting ‘a jack’

Number of favorable outcomes
^{P(B) =} Total number of possible outcome

=1/56

=34/56

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

The probability to have the same number on the two dice is the probability that the second die gives the same number as the first die (whatever this number is). So it is equal to the probability of getting a given number with one die, so it is 16

Now the reduced sample space is of size = 6+(62)=6+15=216+(62)=6+15=21.

Though the sample space is reduced from 3636 to 2121, the probability of getting the same number on both dice is 136136, and the probability of getting different number on both the dice is 236236.

Since we have 66 possibilities of getting same number on both the dice, the required probability is 636=16

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

Variance: The average squared deviation from the mean of the given data set is known as the variance. This measure of dispersion checks the spread of the data about the mean. Standard Deviation: The square root of the variance gives the standard deviation. Thus, the standard deviation also measures the variation of the data about the mean.

Range: Given a data set, the range can be defined as the difference between the maximum value and the minimum value.

Variance: The average squared deviation from the mean of the given data set is known as the variance. This measure of dispersion checks the spread of the data about the mean.

Standard Deviation: The square root of the variance gives the standard deviation Thus, the standard deviation also measures the variation of the data about the mean.

Mean Deviation: The mean devialtion gives the average of the data's absolute deviation about the central points. These central points could be the mean, median, or mode.

Quartile Deviation: quartilr deviation can be defined as half of the difference between the third quartile and the first quartile in a given data set.

Standard deviation has become the most widely used measure of dispersion of data. A measure of dispersion can, in the true sense, be regarded as the proper measure of dispersion if the measure is based on the deviations between all pairs of data. Standard deviation is, however, not based on the deviations between all pairs of data.