Question

What does “distribution sampling” reveal to the researcher. Give an example of how “distribution sampling” is used in a real-world scenario. Write a one to two (1–2) page short paper in which you answer the questions about distribution sampling

Answer 1

* Sampling Distribution. *

In Statistics, we generally draw a sample for investigate the population, to investigate the population we have to compute the Statistic based on sample. Statistic is the function of sample observation. There are infinitely many Statistic for example mean, median, mode etc these all are the Statistic but we need best or sufficient Statistic here mean X bar is unbiased estimate of population mean.

Here, the distribution of Statistic is call as sampling Distribution.

Definition: The sampling Distribution is the probability distribution of Statistic obtain by the sample that we drawn from perticular population. Thus the probability distribution of Statistic is called as sampling Distribution.

Note : Each sample has its own sample mean and the distribution of the sample means is known as the sample distribution.

Assumptions made in sampling Distribution :

One of the important assumptions is that the samples set or the original population have the normal distribution, however, because the samplings distribution involved the multiple sets it need not be Normal distribution.

Now here, sampling distribution is a theoretical distribution of a sample statistic. It is a model of a distribution of samples, like the population distribution, except that the scores are not raw scores, but statistics. It is a thought experiment. "What would the world be like if a person repeatedly took samples of size N from the population distribution and computed a particular statistic each time?" The resulting distribution of statistics is called the sampling distribution of that statistic.

How does the sampling Distribution is used in research ?

In Research are the the researchers were uses the sample to draw the inference about the population. On before drawing the inference about the population they have to draw the samples and compute the Statistic. Here they uses probability distribution on the Statistic to draw the inference about the population.

Real life sichuation where we use the sampling Distribution.

A rowing team consists of four rowers who weigh 152, 156, 160, and 164 pounds. Find all possible random samples with replacement of size two and compute the sample mean for each one. Use them to find the probability distribution, the mean, and the standard deviation of the sample mean X−−.X-.

Solution

The following table shows all possible samples with replacement of size two, along with the mean of each:

Sample	Mean		Sample	Mean		Sample	Mean		Sample	Mean
152, 152	152		156, 152	154		160, 152	156		164, 152	158
152, 156	154		156, 156	156		160, 156	158		164, 156	160
152, 160	156		156, 160	158		160, 160	160		164, 160	162
152, 164	158		156, 164	160		160, 164	162		164, 164	164

The table shows that there are seven possible values of the sample mean X−−.X-. The value x−−=152x-=152 happens only one way (the rower weighing 152 pounds must be selected both times), as does the value x−−=164x-=164, but the other values happen more than one way, hence are more likely to be observed than 152 and 164 are. Since the 16 samples are equally likely, we obtain the probability distribution of the sample mean just by counting:

x−−P(x−−)152116154216156316158416160316162216164116x-152154156158160162164P(x-)116216316416316216116

Now we apply the formulas from Section 4.2.2 "The Mean and Standard Deviation of a Discrete Random Variable" in Chapter 4 "Discrete Random Variables" for the mean and standard deviation of a discrete random variable to X−−.X-. For μX−−μX- we obtain.

μX−−=Σx−− P(x−−)=152(116)+154(216)+156(316)+158(416)+160(316)+162(216)+164(116)=158μX-=Σx- P(x-)=152(116)+154(216)+156(316)+158(416)+160(316)+162(216)+164(116)=158

For σX−−σX- we first compute Σx−−2P(x−−)Σx-2P(x-):

1522(116)+1542(216)+1562(316)+1582(416)+1602(316)+1622(216)+1642(116)1522(116)+1542(216)+1562(316)+1582(416)+1602(316)+1622(216)+1642(116)

which is 24,974, so that

Here answer is √10

Here how sampling Distribution is used in real life sichuation.

Thank you.