Basics of Probability and Statistics Week 1 Challenge

QUESTION1

ANS:

Explanation

In Statistics the term sampling refers to selection of a part of aggregate statistical data for the purpose of obtaining relevant information about the whole. The aggregate or whole of statistical information on a particular character of all the members covered by the investigation is called ‘population’ or ‘universe’. The selected part of the population which is used to obtain the characteristics of the population or universe is termed as ‘sample’. The population is taken to be made of individual units or members, and some of the units are included in the sample. Total number of units of the population is called population size, and that of the sample is called sample size. Population and sample can be finite or infinite and similarly they can be existent or hypothetical

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 between population variance and sample variance

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

Due to this, the value of variance calculated from sample data is higher than the value that could have been found out by using population data. The logic of doing that is to compensate our lack of information about the population data. It is impossible to find out variance of heights in human beings, for our absolute lack of information about heights of all living human beings, not to talk of the future. Even if we take one moderate example, like population data on heights of all the living men in US it is physically possible, but the cost and time involved in this would defeat the purpose of its calculation. This is the reason sample data are taken for most of the statistical purposes, and this is accompanied by lack of information about majority of data. In order to compensate this, the value of variance and standard deviation, which is squared root of variance are higher in case of sample data than variance from population data

Summary: Population variance refers to the value of variance that is calculated from population data, and sample variance is the variance calculated from sample data. Due to this value of denominator in the formula for variance in case of sample data is ‘n-1’, and it is ‘n’ for population data. As a result both variance and standard deviation derived from sample data are more than those found out from population data.

QUESTION2

ANS:Difference between stratified and clustered sampling

Comparison Chart

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 'cluster'.
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
Bifurcation	Imposed by the researcher	Naturally occurring groups
Objective	To increase precision and representation.	To reduce cost and improve efficiency.

conclusion

To end up the discussion, we can say that a preferable situation for stratified sampling is when the identicalness within an individual stratum and the strata mean to vary from each other. On the other hand, the standard situation for cluster sampling is when the diversity within clusters and the cluster should not vary from each other.

Further, sampling errors can be reduced in stratified sampling if between-group differences among strata are increased, whereas the between-group differences among clusters should be minimised to reduce sampling errors in cluster sampling.

QUESTION3

ANS:

Suppose the size of the population is denoted by ‘n’ then the sample size of that population is denoted by n -1. Let us take a look of population data sets and sample data sets in detail. It includes all the elements from the data set and measurable characteristics of the population such as mean and standard deviation are known as a parameter.

QUESTION4

ANS:since there are a total of 4 jacks in a deck of 52, the probability of the getting the first jack is 4:52 = 4/52 = 1/13
.
Now, after the first jack is removed, there are only 3 jacks left in a deck of 51 so the probability of the second card drawn is:
3:51 = 3/51 = 1/17
.
Therefore the probability is:
1/13 * 1/17 = 0.0045

QUESTION5

ANS:

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.

QUESTION6

ANS:

Variance, and its square root standard deviation, measure how “wide” or “spread out” a data distribution is. We begin by using the formula definitions; they are slightly different for populations and samples.

1. Population Formulae :

Variance :

(3.3)

where is the size of the population, is the mean of the population and is an individual value from the population.

Standard Deviation :

The standard deviation, , is a population parameter, we will learn about how to make inferences about population parameters using statistics from samples.

2. Sample Formulae :

Variance :

where = sample size (number of data points), = degrees of freedom for the given sample, and is a data value.

Standard Deviation :

Equations (3.3) and (3.4) are the definitions of variance as the second moment about the mean; you need to determine the means ( or ) before you can compute variance with those formulae. They are algebraically equivalent to a “short cut” formula that allow you to compute the variance directly from sums and sums of squares of the data without computing the mean first. For the sample standard deviation (the useful one) the short cut formula is

At this point you should figure out how to compute , and on your calculator for a given set of data.