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Part 1: Statistical Samples and Populations
In Part 1 of your report, you will again be addressing some fundamental questions asked by
your client to help them understand statistical samples and populations. Address the following
questions:
What is a statistical sample? How are statisticians sure that samples are representative of a
population?
The statistical sample refers to the small group from the whole statistical population having the same
features (Kenton, 2022). Statistical sampling applies to a large population whereby the chosen subset,
will be used to conclude the whole population. This means that statisticians ensure that the sample
statistics contain the “same characteristics” of the targeted whole population to conclude the
research subject. For instance, if the whole target population are 100 Mercedez cars, then the sample
has to be 5 Mercedez car of the same characteristics as the whole population. Shortly, the statistical
sample means selecting a few observable sample sizes out of the same total sample size.
What is an example of a statistical sample for a population?
For instance, if I want to compare the performance and the durability of the laptops given by Kepler
and laptops bought by students themselves, taking the whole Kepler community as my sample size
will take much time. But since Kepler provides the same kind of laptop throughout (Intel Celeron
laptops with the 4.00 GB installed RAM) then, I may take like 20 students as my statistical sample and
select like 10 students who bought their laptops. Then, I may use research methodologies like
questionnaires to formulate my problem statement or conclude which of the two randomly selected
samples possesses the higher performance and a more durable gadget.
What is the importance of random sampling? How can bias be built into non-random
samples?
Firstly, random sampling saves time that would have been used to study the entire population.
Referring to the example above, the time is saved by questioning only 30 students out of more than
400 Kepler students at the Kepler. Furthermore, random sampling gives an equal opportunity for the
population unit to be chosen. For instance, if out of 100 Mercedes Benz cars I choose 5, a statistician
has ensured that all of the 5 sample cars have the distinctive characteristic to ensure that every quality
is tested to conclude the degree of performance for the unique selling proposition. Considering the
example of the Mercedes cars, it is evident that sample sampling leaves the rest of the cars with the
original quality properties (Flexbooks, 2022). Before cars are sold to the public, the company tests the
prototypes with some end-users. The characteristic of equal chances of samples to be selected gives
the third benefit of sample statistics: Unbiased. Bias in sampling occurs when a certain sample is
either “over-represented” or “under-represented” as shown in the explanations below (Flexbooks,
2022). For instance, if I want to compare the Kepler-given laptops of the 2022 cohort following
gender, taking the sample representative of 10 students will balance the number of both genders (
girls and 5 boys). Therefore, the end - results will be reasonable since I assessed them equally.
On the other hand, non-random sampling may lead to bias since one of the sample sizes may be over-
selected or under-selected. Considering the same example of comparing the Kepler-given laptops of
the 2022 cohort following gender, taking the whole cohort of 200 students might result in an unequal
number of boys and girls. As a result, the results will be biased (unequally considered) and hence
won’t result in the possible desirable estimation.
How can a bad sample detract from the accuracy of a statistical measure? Provide an
example.
The bad sample detracts from the statistical measure accuracy by resulting in bias and statistical error.
For example, question private school teachers if they think private schools have more quality teaching
resources compared to public schools. First of all, the answers that will be obtained will be biased since
the private school teachers will have a higher probability of agreeing that their schools are better than
the public schools. Thus, to remove the bias, it is important for the statistician by questioning the same
number of teachers from both private teaching institutions and public teaching institutions.
We can also view the bad sample by not selecting the sample from the same characteristics of the
entire population. Referring to the example about comparing the resources quantity and the quality of
private and public schools, instead of asking teachers, the statistician decides to ask students. The
information that will be obtained won’t be relevant since students are not aware enough of the exact
resources per the number and names available enough to rate the quality. At the end of the day, the
statistician may face a sampling error. Hayes (2022) maintains that the sampling error occurs when a
person performing statistics does not consider the sample size from the targeted entire population.
The probability summation of events is always equal to 1. Therefore the probability
of the sample greater than 9 to occur will be:
Probability = 1-0.93 549
Therefore, the probability that the sample mean is more than 9% is 6. 45 %
A random sample of
16 countries
In the list of random 16 countries, the Mean will be 6.73 and the Standard deviation
will be 2.99 but n will be 16.
Then the z-score will be:
z-score = ( 9 - 6.73)/ ((2.99/16^1/2))
= 3.037 whereby the z-score as on the standard normal distribution table is 0.
Probability:
P ( x>9) = 1-0.
Therefore, the probability that the sample mean is more than 9% of 16 countries is
A random sample of
32 countries
Given:
Mean (μ) = 6.
Standard deviation (α) = 2.
Number of countries/sample size (n) = 32.
Observed value (x) = 9
Then:
z-score = (9-6.73)/((2.99/32^1/2))
= 4.29 which is equal to 0.999 on the standard value of the normal
distribution table.
Probability:
P(x>9) = (1-0.999) *100%
Therefore, the probability that the sample size is more than 9% of 32 countries is
Taking a random sample of 36 countries, determine the probability that the country’s
domestic government health spending (as a percentage of total government spending) is
above 8%, assuming it is a normal distribution.
The mean and the standard deviation are shown in the screenshot below:
Observed value (x) = 8
Sample size = 36
Therefore,
Z score = (8-10.63) / ((5.702/36^1/2))
= - 2.767 which on the standard values of the normal distribution is equal to 0.
Probability:
P (x>8) = (1-0.0028) *100%
Therefore, the probability that the 36 country’s domestic government spending is above 8% is 99.72%
Taking a random sample of 36 countries, determine the probability that the proportion of
health spending (by percentage of GDP) is greater than 10%, assuming it is a normal
distribution.
Given:
Mean (μ) = 10.
SD= 2.
Sample size (n) = 36
Observed value (x) = 10
Then:
z-score= ( 10 - 6.73)/ ((2.99/36^1/2))
= 6. 56 which on the standard z-scores read 1.
Probability:
P (x>10) = (1-1) *100%
that enough testing was not effectively done to meet the normality. Also, we may confirm the
asymmetrical shape formed by 46 countries by examining the histogram from the descriptive data
which shows the characteristics of the observable sample. All in all, enough testing such as hypothesis
testing, considering equal chances per each country either European or not, and testing the normality
must be considered to avoid overestimating the sales forecasting.
References
Flexbooks. (2022, August 12). 13.13 Sampling methods. From flexbooks.ck12.org:
https://flexbooks.ck12.org/cbook/ck- 12 - algebra-i-
concepts/section/13.13/primary/lesson/sampling-methods-alg-i/
Flexbooks. (2022, August 15). 2.2 Population vs. Sample. From flexbooks.ck12.org:
https://flexbooks.ck12.org/cbook/ck- 12 - probability-and-statistics-
concepts/section/2.2/primary/lesson/population-v.s.-sample-pst/
Hayes, A. (2022, September 06). Sampling Error. From www.investopedia.com/:
https://www.investopedia.com/terms/s/samplingerror.asp
Kenton, W. (2022, July 1). Sample. From www.investopedia.com:
https://www.investopedia.com/terms/s/sample.asp
Saylor. (n.d.). 16.3 Forecasting. From saylordotorg.github.io:
https://saylordotorg.github.io/text_principles-of-marketing-v2.0/s19- 03 - forecasting.html
SNHU. (n.d.). Sales Forecast Process. From learn.snhu.edu:
https://learn.snhu.edu/content/enforced/841542-MAT- 20677 - OL-DAC-
CFA.21CFA/Competency%20Files/MAT-
20677%20Sales%20Forecasting%20Process.pdf?_&d2lSessionVal=81PEiZhSf5fxiYXm
yyvEH20Z2&ou=
z-table. (n.d.). Z Table. From www.z-table.com: http://www.z-table.com/
