Analyzing Central Limit Theorem in Business: Homework Solution

This assignment provides a comprehensive analysis of the Central Limit Theorem (CLT) and its applications in business. It includes a discussion of the CLT, explaining its principles and significance in statistical analysis. The solution demonstrates how to determine probabilities related to sample means, assess differences in income using Z-tests, and evaluate the effectiveness of new processes compared to old ones. The assignment incorporates real-world examples from the clothing industry and TV tube manufacturing, using statistical tests to draw meaningful conclusions. It also includes interpretations of the Z-test values and the significance of the findings, supported by relevant references. The assignment helps students understand how to apply statistical methods to make informed business decisions.

Quantitative
methods in
business

Central Limit Theorem................................................................................................................3
a. Determining probability or chances where average time would be lesser than 9.5 minutes...3
b. Indicating that income from the self-employment in clothing segment is significantly
different........................................................................................................................................4
c. Identifying that new process is better than old one..................................................................4

Central Limit Theorem
With respect to the study of the probability theory, this theorem states that distribution of
the sample means as approximate normal distribution because the size of the sample becomes as
large, assuming that each and every sample are reflected as identical in the size, regardless of the
population within the distribution sample (Kwak and Kim, 2017). It is the statistical theory
which states that the provided sample from the population with finite variance level, average of
all the samples from the same kind of population would be approximately resulted as equal to
mean of population. Furthermore, all samples would follow an anticipated pattern of normal
distribution, along with all the variances counted as equal to population variance divided by each
of the sample size.
In accordance to central Limit theorem, mean of sample of the data would be seen as
closer to mean of an overall population in the question, as sample size increase, actual
distribution of data also increases (Hairer and Shen, 2017). In the other words, data is resulted as
accurate whether distribution is aberrant or a normal. As per general rule, size of sample equates
to or is greater than 30 which deemed as sufficient for CLT to holding. It means that sample
distribution is fairly seen as normally distributed. Thus, more samples with one task, more the
graphed results shape for the normal distribution. These theorem exhibit phenomena within
which average of sample mean and the standard deviation is seen as extremely useful for
predicting the characteristics of the population accurately.
Z-test distribution- It refers to the statistical test that is been used for determining that
whether the two population mean are resulted as different when variances are been known and
size of sample is large (Sirignano and Spiliopoulos, 2020).
a. Determining probability or chances where average time would be lesser than 9.5 minutes
Particular
s Time
μx 10.5
σx 3
n 50
Mean 9.5

Find P( x
< 9.5)
Z=
Mean-μx/σx/Square
root of n
9.5-10.5/3/square root
of 50
Z= -2.36
Interpretation- From the above table it has been analyzed that value of Z resulted as -
2.36 by subtracting the sample mean from mean and dividing it by standard deviation and the
square root of the sample (Dinur and Livni Navon, 2017). This value states that the average
chance for which the time resulted as less than 9.5 minutes is -2.36 as per the z table distribution.
b. Indicating that income from the self-employment in clothing segment is significantly different
Particular
s Time
μx 15000
σx 975
n 169
Mean 14500
Z= Mean-μx/σx/Square root of n
14500-15000/975/square
root of 169
Z= -0.26
Interpretation- The above table depicts that the z value in respect of self- employment
with clothing industry is seen as significantly different valuing as -0.26 (Ross and Willson,
2017).
c. Identifying that new process is better than old one
Particulars Time
Mean tube 4700
Standard
deviation 1460
New tube 100

sample mean
Life 5000 hrs
Level of
significance 1%
Ho: μ= 4700
Ha: μ > 4700
(5000)=
(5000-4700)/[1460/
sqrt(100)] =
3000/146
0
2.05479
5
P-value: P(+>2.05 with df=99)
Since significance or p-value resulted as greater than 1%, therefore null hypotheses is
accepted.
Thus, new process does not seems as an improvement over the old process.
Interpretation- The above table shows that the significance or P-value resulted as higher
than 1% which clearly means that null hypotheses would be accepted and the alternative one will
be rejected. This in turn means that the new process is not considered as improved one over older
processes (Kosambi, 2016). Therefore, manufacturer of TV tubes must not opt for new processes
over the old one as it will not result to success for the firm in future and in present.

REFERENCES
Books and journal
Dinur, I. and Livni Navon, I., 2017. Exponentially small soundness for the direct product z-test.
In 32nd Computational Complexity Conference (CCC 2017). Schloss Dagstuhl-Leibniz-Zentrum
fuer Informatik.
Hairer, M. and Shen, H., 2017. A central limit theorem for the KPZ equation. The Annals of
Probability. 45(6B). pp.4167-4221.
Kosambi, D. D., 2016. A bivariate extension of Fisher’s z-test. In DD Kosambi (pp. 87-91).
Springer, New Delhi.
Kwak, S. G. and Kim, J. H., 2017. Central limit theorem: the cornerstone of modern
statistics. Korean journal of anesthesiology. 70(2). p.144.
Ross, A. and Willson, V. L., 2017. One-sample T-test. In Basic and Advanced Statistical
Tests (pp. 9-12). Brill Sense.
Sirignano, J. and Spiliopoulos, K., 2020. Mean field analysis of neural networks: A central limit
theorem. Stochastic Processes and their Applications. 130(3). pp.1820-1852.