Sampling Distributions Assignment help




A Sampling Distribution or finite-sample distribution is the distribution of a given statistic based on a random sample of size n. It may be considered as the distribution of the statistic for all possible samples from the same population of a given size. The sampling distribution depends on the underlying distribution of the population, the statistic being considered, and the sample size used. The sampling distribution is frequently opposed to the asymptotic distribution, which corresponds to the limit case n → ∞. For example, consider a normal population with mean μ and variance σ². Assume we repeatedly take samples of a given size from this population and calculate the arithmetic mean for each sample — this statistic is called the sample mean. Each sample has its own average value, and the distribution of these averages is called the “sampling distribution of the sample mean”. This distribution is normal since the underlying population is normal.

Types of Sampling Distribution:-

The sampling distribution is an allocation of a sample statistic. While the notion of a distribution of a set of numbers is instinctive for most students, the concept of a distribution of a set of statistics is not. So distributions will be reviewed previous to the sampling distribution is thrash out. The solution distribution of statistics is called the sampling distribution of that statistic. Sampling distribution is a probability distribution so total probability should be equal to one.Let us see types of distribution.

 

Following are the types of sampling distribution:

Random and simple random Sampling:

     Random sampling are an important types which has all groups must be proportionately represented. Its sampling is together with representative and proportionate. For Example members of the population are preferred in such a way that all have an equivalent chance to be measured.

      Simple random sampling is consists of n elements starting from the population preferred in such a way that all set of n those has an equivalent change of being the sample really selected. Both types are sampling distribution.

Systematic Sampling:

      Systematic sampling is every kth member of the population is sampled. If a sample of volume s is to be taken from a population of size n, and then every n/s member of the population is tested. The initial point is selected at random. If we want to test a 200-strong sample from a population of 4000, we test every 4000/200= every 20th member. We use random numbers to find out the starting point.

Stratified Sampling:

       A stratified sample is through up of different 'level' of the population, for example, selecting samples from different age groups. The sample size is proportional to the size of the 'level'. This is shown in the following equation:

Sample size for each level = size of level / size of population * size of whole sample

Cluster Sampling:

      A population is separated keen on clusters and a small number of these frequently randomly selected, clusters sampling be exhaustively sampled. Exhaustively means allowing for every part of elements.

Convenience Sampling:

     Convenience sampling is completed as convenient, frequently allowing the element to prefer whether or not it is sampled. Convenience sampling is the simplest and potentially most unsafe. Often good results can be getting, but maybe just as frequently the data set may be critically biased. These are the types of sampling distribution.

Introduction to Sampling distribution of :


                      Draw the possible sample outcomes of size "n", for a given finite population of size N. Then the total number of all possible samples which is having the  same size n which can be drawn from the population is given by Ncn  = N! / (n! (N – n)!) (N!) / (n! (N - n)!) = k. for each of these sample using the sample data x1, x2, x3….x n by S = S(x1, x2, x3….x n).

                                                        Sample Numbers            1       2      3. .  . . . . . . . .  k s

                                          Statistics(S)                     S1      S2    S3                            S k

          It is called sampling distribution.

Sample Problem for Sampling Distributions:

sampling distributions problem 1:   

Sol:  Sampling distribution of means (S.D.M)img is the probability distribution of img .

Finite population:

              Let us consider a finite population of the sample size N with mean µ and standard deviation σ. Draw all possible samples of size from this population without replacement. Let µ img  and  σimg indicate the mean and standard deviation of the sampling distribution of means. Suppose  the value N > n. then

                                                 µimg µ

.                                               σ imgσ/(sqrt(N))  sqrt((N - n)/(N + n))

        Here the finite correlation factor issqrt((N - n)/(N + n))

Sampling Distribution of Mean:

sampling distributions problem 2:

             If we drawn the samples from an infinite population or sampling is done with replacement then

                                                                          µimg = µ and

.                                                                        σ imgσ/(sqrt(N))

                            S . E (standard error) of mean σimgσ/(sqrt(n)), which measure the reliability of the mean as an estimate of the population of mean µ 

                                                            Standard sample mean, Z = (img + µ) / σ √n

Sampling Distribution of proportions:

                                     Let us take p be the set of occurrence of an particular event (success) and q = 1 - p is the probability of non - occurrence (failure)Draw all possible sample size n from an infinite population. Now we have to calculate the proportion P of success for al samples. Then the mean µp and variance σp2 of the sampling distribution of proportions are given by

                                                        µp = P

                                                       σp2   =(p q)/(n) = (p(1 - p))/(n)

      When the population is binomially distributed, the sampling distribution of the proportion is normally distributed if n is large. For finite population (with replacement) of size N

                                         µp = P   and 

                                         σp2 = (p q)/(n) ( (N+n)/(N-1) )

Small Sample Distribution:-->

Introduction of small sample distribution:

The probability theory and statistics use the sample two distributions. They are binomial and Poisson distribution. The basic and small sample distribution is the binomial distribution, and it is also small distribution compared with Poisson distribution and normal distribution etc. The number of successes for discrete probability distribution in sequence of ‘n’ independent event is called as binomial distribution. Let the probability of getting a success in a single trial be p and that of getting a failure be q so that p+q = 1.

                                     p(X =r) = ncr pr q(n-r) where q =(1-p)

Small Sample Distribution (binomial Distribution)

Properties of sample Binomial Distribution

• The experiment has n repeated trials.

• Each trial has the possible outcomes. One is success and another one is failure.

• In this trials are independent.

• Mean = n * P.

• Variance = n * P * (1 – P).

• Standard Deviation =√ [n * P * (1 – P)].


B(x; n, P) = Binomial Probability.

   r = number of successes

   n = number of trials

   P = Probability of success

  nCr = Number of combinations of n trials, x is success.

Small Sample Distribution Problems

Sample problem 1: A die is tossed 6 times. What is the Probability of getting exactly 2 fours?

Sol:

Here n = 6, x = 2, probability of success on a single trial = 1/ 6 = 0.167.

Then p = 0.167,

p + q =1

p = 1-q

Formula P(b) = ncr pr q(n-r) or q =(1-p)

Therefore, The binomial probability is,

b( 2; 6, 0.167 ) = 6C2 × ( 0.167 )2 × ( 1 – 0.167)6 – 2

                           = ( 6! / 2! × (6-2)!) × 0.0279 × ( 0.833)4

                           = (6! / 2! × 4!) × 0.0279 × 0.481

                           = 15 × 0.0279 × 0.481

b( 2; 6, 0.167 ) = 0.201. Answer.

Sample problem 2: A die is tossed 2 times. What is the Probability of getting exactly 1sixes?

Sol:

Here n = 2, x = 1, probability of success on a single trial = 1/ 2 = 0.5.

Then p = 0.5,

p + q =1

p = 1-q

Formula P(b) = ncr pr q(n-r) or q =(1-p)

Therefore, The binomial probability is,

b( 1; 2, 0.5 ) = 2C1 × ( 0.5 )1 × ( 1 – 0.5)2 –1

                       = ( 2! / 1! × (2-1)!) × 0.5 × (0.5)

                       = (2! / 1! × 1!) × 0.5 × 0.5

                       = 2 × 0.5 × 0.5

b( 1; 2, 0.5 ) = 0.5. Answer.

Practice problems for binomial distribution

Sample Practice problems:

1) A die is tossed 4 times. What is the Probability of getting exactly 3 fours?

2) A die is tossed 7 times. What is the Probability of getting exactly 5 twos?

Answers:

1) 0.048

2) 0.0015

Sample Distribution Plan:-

Introduction to sample distribution plan:

                   The sampling distribution includes the normal probability distribution and the binomial probability distribution. The normal distribution is also called as the Gaussian distribution. In normal distribution the mean is μ and the variance is σ^2 . Binomial distribution is the close approximation to the normal distribution. The normal distribution is referred as the limited form for the Poisson distribution probability. This article has the details about the sampling distribution plan.

Formula Used for Sampling Distribution Plan:

The formula used for the normal random variables are

 Z = (X- μ) /σ

Where X is the normal probability with mean μ and the variance is σ^2 , σ is the standard deviation.

The formula used to find the binomial distribution is,

P (X = x) = n Cx px (1-p) (n-x)

Where n is the number of possibilities for the event and x is the required probability for the given random variables.

Examples for the Sampling Distribution Plan:

Example 1 for the sampling distribution plan:

 If X is normally distributed the mean value is 1 and its standard deviation is 6. Determine the value of P (0 ≤ X ≤ 2).

Solution:

The given mean value is 1 and the standard deviation is 6.

Z = (X- μ)/ σ

When X = 0, Z = (0- 1)/ 6

                       = -1/6

                       = -0.17

When X = 2, Z = (2- 1)/ 6

                       = 1/6

                       = 0.17

Therefore,

P (0 ≤ X ≤ 2) = P (-0.17 < Z < 0.17)

P (0 ≤ X ≤ 2) = P (0 < Z < 0.17) + P (0 < Z < 0.17) (due to symmetry property)

P (0 ≤ X ≤ 2) = 2 (0.5675- 0.5)

P (0 ≤ X ≤ 2) = 2(0.0675)

P (0 ≤ X ≤ 2) = 0.135

The value for P (0 ≤ X ≤ 2) is 0.135.

Example 2 for the sampling distribution plan:    

           A coin is flipped 4 times. Determine the probability to getting the tail exactly 2 times.

Solution:

The coin is flipped 15 times. So n = 4 and x = 2. The probability for getting a tail in a trial is p = 1/2

The formula used to find the binomial distribution is,

P (X = x) = n Cx px (1-p) (n-x)

P (X = 2) = 4 C2 (1/2 )2 (1- 1/2 ) (4-2)

P (X = 2) = 6 (1/2 )2(1/2 ) 2

P (X = 2) = 6 (0. 25) (0.25)

P (X = 2) = 0.375

The probability to get the tail exactly 2 times is 0.375.

Sampling Distribution of Proportion:-->

Introduction to Sampling and Sampling Distributions:

Often in food stalls, the shoppers often taste a small piece of an item and based on their experience with the small piece they decide to buy or not to buy. This is an example of sampling. This is done in many factories as well. Consider a tyre manufacturing factory. The quality inspection engineers pull out a few manufactured tyres and test the tyres thoroughly and in that process these tyres are destroyed.

Note in the previous two cases why was the sampling done? If the shoppers in the food stall had tasted the entire quantity available, there would have been nothing left for sale. Similarly if the quality inspector had tested all the tyres, all the tyres would have been destroyed and there would have been nothing available to sell. So sampling for inspection was unavoidable in these cases.

Consider another case where you want to study about the average income of graduates in a particular region. If we try to get information from all the graduates in the region, the process will be time consuming and would be elaborate. However, if we collect data from a representative group of people, the same can be done quickly and easily

In the above example we deduce information about a bigger set based on a sample. The bigger set is called the population. We infer the data of the population based on the sample data. In the above cases, the total amount of the food in the food stall, total tyres produced by the factory, total graduates in the region are the population. The total tyres produced by

If the above samples are taken repeatedly and the mean of each sample is plotted is plotted against its probability then the distribution obtained is called sampling distribution.

Sampling Distribution of Proportions:

A sampling distribution described above can be partially described by the mean and the standard deviation. If we collect samples repeatedly from a population and calculate the mean, there will be difference in means for different samples. The means will be different. This is because each sample has different sampling elements from the population. This variation is called the standard deviation of the distribution of sample means or simply as standard error of the means. Similarly, the standard deviation of each sample might vary and this variation in the standard deviation is called the standard deviation of the sampling distribution or simply as standard errors.

The aim of taking samples is to estimate the population characteristics like mean and standard deviations. If we take samples repeatedly then the values of mean and standard deviation for each sample will be different. If so, which one will we consider as an accurate representation of the population data?  The standard error tells us how reliable it will be if we predict the population values from the sample statistics. There is an important theorem correlating the characteristics of the sample and the population and this is called the central limit theorem

Central Limit Theorem says the mean of the sampling distribution will be equal to the population mean regardless of the sample size. The sampling distribution of the means will approach normality as the sample size increases. This theorem helps to make inference about the population without knowing much about the population or the distribution of population.

This means the sampling distribution means will be the population means and the standard deviation of the population is accurately estimated if we use a large sample size. This is illustrated in the figure below. As we can see form the graph, as the sample size increase the mean is the same but the standard deviation decreases.

img

The process of evaluating the population data from the sample is called estimation. Central limit theorem is the basis on which the theories of estimations have been propounded. The theory of estimation helps us to find the population values of mean and standard deviation form the sample values.

Statisticians often use a sample to estimate the proportion of occurrences in a population. For example if we want to measure the unemployment rate or the proportion of unemployed people in the population. When we make try to estimate the proportion of a population from the sample we call it as a estimate of the population proportional and the sampling distribution is the sampling distribution of the proportion.

Exercises on Sampling Distribution:

Q:1 A machine is supposed to fill on an average 125 grams of a liquid in a bottle with a standard deviation of 20 grams. A random quality inspection shows a mean of 130. The inspector concludes that the sample is wrong. Is he correct?

Ans: No. The sample data need not accurately represent the mean. So just because the sample value was 130 grams does not mean it is a wrong sample.

Q:2 A sample of patients with tooth diseases wanted to be carried out regarding their brushing and eating habits. The statistician approaches a group of dentists and asks them to submit data. Each dentist takes data from 50 patients and submits the average value to the statistician. The statistician draws inference based on this. Was this a sampling of the patients?

Ans: No. The statistician used the data from the dentists, which was a mean of the 50 patients and not individual patient. So the statistician had plotted a sampling distribution and not a sample value.

Prob 3: What sample size will give you mean value equal to the population mean and without any error?

Ans: The sample size should be the same as population. This means that we need to carry out the data collection for the entire population.


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