University of Tampa Probability Density Function Discussion

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1. General Introduction
The PDF (probability density function) plays a critical role among various data sets. The Central
Limit Theorem (CLM) nevertheless enumerates that the sample mean (x̄) distribution assumes a
more spread progressively normally with the increase in the sample size (n). Ordinarily, a sample
size of not less than n=30 is preferred to gain a normally spread sample mean distribution. To
demonstrate the Central Limit Theorem, one can envision a dice with equally assigned numbers
from one to six. If we estimate two dice, then we start seeing a tapering on the lower and higher
figures. The PDF sample mean assumes a near-normal spread when the dice is averaged at ten.
The statistical principle occupies a central place in several research applications, and it is, at
times, considered practical to obtain samples in the place of scrutinizing the population. Besides,
we can derive statistical theorems, including the equality of the sample mean with the sample
mean through a normally spread sample.
The aim of this lab is to prove the Central Limit Thermos.
2. Procedure and Statistical Principle
On the outset, there were six distinct excel pages created as a worksheet. They comprised
“Uniform Random Number 10”, “Exponential Random Number 10”, “Histogram 10”, “Uniform
Random Number 30”, “Exponential random Number 30,” and “Histogram 30” The
categorization helps organize and label data. Ten columns of 110 evenly distributed random
numbers beginning from 0 to 1 were designed with the RAND function, which is an exclusively
regularly distributed data set similar to the dice comparison cited earlier. In the next step, data
was applied to create our increasingly spread data set on the second page based on the equation
below.
�(�) = ��^(−��) for x>0
The lambda value of .1 was arrived through the division as indicated (applying -LN(1-x)/.1 in
Excel). The function assists in creating our PDF based on the exponential equation expressed by
the e base and -x� power. Several other possibilities could illustrate similar statistical concepts.
A histogram of the data was created by first deciding the number of bins required. The counting
of the number of data points (n) was then done and the square root of the figures taken to
determine the number of bins. For each data set, the number of bins was established.
Additionally, the bin size was obtained from every data set. The calculation was achieved by
dividing the bin range by adding bin size to the preceding bin to achieve the desired number of
bins. Finally, the descriptive statistics add-ons were applied to develop a histogram. The original
data was gathered, and histogram bins obtained before formatting of the plot area to form a
histogram for the values in histogram 10 tab. The histogram is illustrated below.
The next step involved the computation of 110 sample mean values (x̄) from the highly
distributed values, which was achieved by computing the average for each row in the exponential
random number 10 tab. Consequently, a histogram was created in the manner of the procedure
adopted above but applying the sample mean in the place of raw data. As illustrated below, the
histogram is expected to show a normal distribution. It was ascertained through creation of a
normality plot. In this regard, the guidelines in the textbook were adopted to plot zi vs xi that
should be more or less linear with a higher concentration near the center of the values. The value
of x was arranged from low to high and the z values calculated following the equation P(Z<=z)
=(j-./n) where j is the integer order of the selected x data To compute these values.
Finally, the same procedure was applied for the above values, but 30 columns were used in the
place of ten in this case.
3. Results and Discussion:
Six different figures were developed through this experiment (3 for n=10 and 3 for n=30 sample
sizes). The results were histograms of the randomly picked numbers, and the corresponding
estimates. Every figure communicated varied information. In total, however, the proof of the
Central Limit Theorem was accomplished.
Initially, the histograms from the primary data showed the lack of uniformity in the set of
numbers that were at the start followed by the equations that were created. This realization
proved that the Central Limit Theorem is applicable for all kinds of PDF’s beside the uniform
PDFs in the introduction.
The Histograms of the estimated values are considered very critical in this aspect of the lab.
They enabled the visualization of the transformation from the exponential curve in the preceding
histograms to a balanced “bell curve” shape. Nevertheless, human sight is deceptive in scaling,
and other plotting aspects could blur our judgment. To verify the normality of the sample mean
histograms, a near-linear spread of the data in the normality plots were observed.
In the end, enhancing the sample size to thirty increased the normality of the sample mean in
accordance with the Central Limit Theorem as earlier predicted. The normality plot had a value
of n=30 was more linear with limited skewness and showed that the histogram was normal.
Experimental errors are expected in this nature of the experiment, and the application of Excel to
create random numbers significantly minimizes bias because they are system generated. In
addition, all the computations were managed from the Excel environment and, as such,
interaction from the cells was expected to be clear. Nevertheless, Excel doesn’t have an
application that calculates all random numbers with the creation of each new cell. This was a
huddle that was jumped by using the copy and pasting the data strictly from the value in each cell
and not the equation. However, it is expected that some information would be lost when creating
histograms. The outlined procedure was followed, but the histogram inherently fails to
incorporate details in the bins. Nonetheless, a second figure for referencing was incorporated in
the normality plots.
4. Conclusion
The steps taken in the lab demonstrate the regularizing power of the Central Limit Theorem. The
procedure was done by obtaining the sample mean from the highly spread data and consequently
developing a normally distributed sample mean sets. Histograms and normality plots were
critical in illustrating this position. Besides, the initial experiment was improved from the initial
status by enhancing the sample size from 10 to 30. Based on the propositions of the Central
Limit Theorem, a bigger sample size improves the normality of the sample means.
As a general rule, this statistical tool is quite useful as it enables the normalization of data
without the difficulty initially experienced. Following the normalization, known statistical
principles like Z-scores and T-scores for additional analysis (i.e. statistical significance and
hypothesis testing)
5. Future Application
As indicated earlier, the Central Limit Theorem is a statistical tool that is used by many
specialists, including scientists and engineers on a regular basis. In the bioengineering field, this
process is used to comprehend non-normal data sets. The totality of the bioengineering field is
premised on applying its tools to enhance the health conditions and provide remedies for
diseases. The process begins with an analysis of the target population. Several companies in Salt
Lake Valley, for instance, specialize in the manufacture of devices that manage cardiovascular
diseases like stents and balloons to expand arteries (as a measure against atherosclerosis and
stenosis). The process is also used to manage implantable cardioverter defibrillators and
pacemakers that manage the heartbeat (and prevent heart attack). Whereas the general public
could suffer from heart ailments, it would be cumbersome to test every individual using the
equipment. There is needed to carefully identify individuals who are a reasonable patient that can
benefit from the devices. To accomplish this, data about the distribution of heart disease with
respect to age. As expected, the process would yield non-uniform results. A search from the
internet for blood pressure and congestive heart failure show a skewness in favor of older
populations. It is a reasonable expectation and helps in inquiries about the mean and standard
deviation, among other statistical computations of the data. Observation of the raw data poses a
challenge in this respect. In this regard, the Central Limit Theorem is used.
A sample of 100 patients can be considered as an example. The patients may present with high
blood pressure from across the country in various hospitals. Although several methodologies
could be applied, the most preferred would be a random selection of the respondents in different
hospitals and documentation of their age, among other factors. The sample size of 100 is
practical considering that it is collected from across the country. Nevertheless, based on the
Central Limit Theorem, there will need to create a normal sample mean distribution. Continuous
collection of more samples will enable the establishment of a normally distributed data set of
sample means as previously ascertained. In this regard, the sample means are normalized and can
enable testing of the probability of its impact on a given age range with respect to high blood
pressure and other heart diseases of interest (If similar tests are performed under different
conditions). With the information thus obtained, statistics could be used to determine the need
for further devise intervention. The example below illustrates the kind of raw data that can be
collected.