Business Statistics: Summary, Hypothesis Testing, and Conclusion

This report provides a summary of the data, hypothesis testing, and conclusion on the willingness for a unisex gym among male and female respondents. It includes calculations of confidence intervals and test statistics, as well as explanations of the appropriate methods to summarize data based on variable types. The report also discusses the structure of a good report that uses statistics and the hypothesis tests used in the report.

Business Statistics
Student Name:
Instructor Name:
Course Number:
19 September 2018

“Section 1 reproduction of the data summaries including details of simple calculations
1a) Summary of the variable ‘do the customers want a unisex gym’ just considering the females “
sample 315
gender Female
Row Labels Count of Should the gym be Unisex?
no 12
yes 31
Grand Total 43
The sample proportion that say yes is;
^p= x
n = 31
43 =0.7209
From the calculation, the sample proportion that say yes is 0.7209 representing 72.09% of the sample.
“ 1b) Summary of the relationship between the variables ‘time on cardio machine’ and ‘time on weight machine’ “
0 10 20 30 40 50 60 70
0
10
20
30
40
50
60
f(x) = − 0.727199268987081 x + 36.2572213957212
R² = 0.780882671949939
Minutes on Weight machine
Minutes on Cardio
Minutes on Weight machine
There is a positive linear relationship between minutes on Cardio and minutes on weight machine.
“1c) Summary that lets you investigate the relationship between the variable ‘does the customer want a unisex
gym’ and ‘gender’ “
sample 315
Count of Should the gym be
Unisex? Column Labels
Row Labels no
ye
s
Grand
Total
Female 12 31 43

Male 48 9 57
Grand Total 60 40 100
sample 315
Count of Should the gym be
Unisex? Column Labels
Row Labels no yes
Grand
Total
Female 27.91%
72.09
% 100.00%
Male 84.21%
15.79
% 100.00%
Grand Total 60.00%
40.00
% 100.00%
A large proportion of female respondents want the gym to be Unisex as compared to that of the males.
“1d)Summary that lets you investigate the relationship between the variable ‘time spent on the cardio machine ’
and ‘gender’ “
Row Labels Average of Minutes on Cardio
StdDev of Minutes on
Cardio
Female 38.02326 14.56265
Male 15.96491 14.80343
Grand Total 25.45000 18.28637
On average, females spend more time on Cardio (M = 38.02, SD = 14.56) as compared to the males (M = 15.96, SD =
14.80).
“Section 2: calculation of confidence intervals and test statistics
2a) Calculation of confidence interval for the proportion of females that prefer a Unisex gym using my allocated
sample”
90% confidence interval for sample proportion for females
Count of Should the gym be
Unisex? Column Labels
Row Labels no
ye
s
Grand
Total
Female 12 31 43
Male 48 9 57
Grand Total 60 40 100
Considering females only, we have;
N = 43,
Sample proportion ^p= x
n = 31
43 =0.7209

Standard error of sample proportion = √ 0.7209(1−0.7209)
43 = √ 0.004679=¿ 0.068404
90% confident the true proportion is between
0.7209 ± 1.645∗0.068404=0.7209 ± 0.1125
Lower limit: 0.7209−0.1125=0.6084
Upper limit: 0.7209+0.1125=0.8334
“2b) Calculation of confidence interval for the proportion of males that prefer a Unisex gym using my allocated
sample”
Note you use the information from section 1c
90% confidence interval for sample proportion for males
Count of Should the gym be
Unisex? Column Labels
Row Labels no
ye
s
Grand
Total
Female 12 31 43
Male 48 9 57
Grand Total 60 40 100
Considering males only, we have;
N = 57,
Sample proportion ^p= x
n = 9
57 =0.1579
Standard error of sample proportion = √ 0.1579(1−0.157 9)
57 = √0.002333=¿ 0.048299
90% confident the true proportion is between
0.1579 ± 1.645∗0.048299=0.7209 ±0.07945
Lower limit: 0. 1579−0.07945=0.07845
Upper limit: 0. 1579+0.07945=0.23735
“2c) Calculation of Test stat for checking if the females that want a unisex gym is above 50%
(Find the sample proportion’s zscore assuming p=0.5) using my allocated sample”
Test stat z= ^p− p
√ p(1−p)
n
assume p=0.5
Count of Should the gym be Unisex? Column Labels

Row Labels no
ye
s
Grand
Total
Female 12 31 43
Male 48 9 57
Grand Total 60 40 100
Considering females only, we have;
N = 43,
Sample proportion ^p= x
n = 31
43 =0.7209
z= ^p− p
√ p(1−p)
n
=
0.7209−0.5
√ 0.5(1−0.5)
43
= 0.2209
0.076249 =2.8971
The computed z-score is greater than the critical z value of 1.96; we therefore reject the
null hypothesis and conclude that the proportion of females who want a unisex gym is
above 50%
“2d) Calculation of Test stat for checking if the males that want a unisex gym is above 50%
(find the z score of proportion assuming p=0.5) using my allocated sample”
Test stat for checking if the males that want a unisex gym is above 50%
Count of Should the gym be Unisex? Column Labels
Row Labels no
ye
s
Grand
Total
Female 12 31 43
Male 48 9 57
Grand Total 60 40 100
Considering males only, we have;
N = 57,
Sample proportion ^p= x
n = 9
57 =0.1579
z= ^p− p
√ p(1−p)
n
=
0.1579−0.5
√ 0.5(1−0.5)
57
= −0.3421
0.066227 =−5.1656
The computed z-score is greater than the critical z value of 1.96; we therefore reject the
null hypothesis and conclude that the proportion of males who want a unisex gym is below
50%.

“Section 3
Evidence I can decide the appropriate method to summarize data based on the nature of data (the variable types)
a) My explanation why the different main findings in the sample report use different methods to summarize data.
“
The sample report sought to understand whether the business should have a separate gym for women only.
The study sampled 100 respondents and the respondents were asked a set of 4 questions (with two qualitative
and 2 quantitative questions). The datasets therefore were as follows;
Variable Type Scale
Gender Qualitative Nominal
Prefer unisex gym Qualitative Nominal
Time on Cardio Quantitative Ratio
Time of Weight machine Quantitative Ratio
In terms of analysis, the authors gave the frequency distribution tables as well as the bar charts for the
nominal scale variables. For the ratio scale variables, the authors computed the measures of central tendency
such as the mean and measures of variation such as the standard deviation.
b) “My summary of an article that discusses gyms and an appropriate numerical summary using my allocated
sample from the section 3 dataset”
The article I sought to compare was based on the average heights.
The link for the article is given as https://ourworldindata.org/human-height.
In the article, the report the average for the German’s was found to be 180.5 cm in the year 1980. The results
from the article is given below;
Using the dataset provided, we found out that average height for the Germans to be 168.67 cm
Q2: Height
Mean 168.67
Standard Error 1.388339
Median 168
Mode 154
Standard 13.88339

Deviation
Sample Variance 192.7486
Kurtosis -1.15196
Skewness 0.064955
Range 50
Minimum 145
Maximum 195
Sum 16867
Count 100
As can be seen, there are slight differences but may be the differences could be as a result of change in time
period.
“Section 4 my discussion of the webpages used to do all the calculation of a hypothesis tests
4a use a webpage to do a hypothesis test of the difference between two proportions using my allocated sample
Using
http://www.socscistatistics.com/tests/ztest/Default2.aspx
“
As can be seen, the proportion of female respondents who would want the gym to have a unisex is significantly
higher than that of the males.
“”“4b) a use a webpage to do a hypothesis test of the difference between two means using my allocated sample

Using
https://www.graphpad.com/quickcalcs/ttest1/?Format=SD
“
“The output is “
The output shows that there is significant difference in the amount of time taken at Cardio for the males and the
females.

“Section 5
Appropriate simple conclusions based on the computer output of hypothesis tests in section 4”
The first part of section 4 sought to find out whether there is significant difference in the proportion of the male
and female respondents advocating for unisex gym. Results showed that a large proportion of female respondents
would want to have a unisex gym as compared to the male respondents.
For part b of section 4, the aim was to check whether there is significant difference in the amount of time taken at
Cardio for the males and the females. Again, results showed that there is indeed a significant difference in the
amount of time taken at Cardio for the males and the females. The females tend to take longer time as compared
to the males.
“Section 6
Demonstrates I can explain the structure of a good report that uses statistics “
The main of this report was to compare the willingness for a unisex gym among the male and female respondents.
The sample report is divided into several sections. The first section sought to provide descriptive summary. The
structure is very clear in the sense that it goes step by step by reporting on the descriptive statistics before going to
the real hypothesis test.
In my understanding the structure of this report is well articulated and well presented.
“Section 7
Demonstrates I can explain the hypothesis tests used in report and hypothesis testing in general“
There are four hypothesis tests used in this report. A hypothesis test is that test that is used to verify a claim, that
is, to find out whether a claim is true or false.
The first hypothesis tested in this report is in relation to the research question; Are more female respondents for a
unisex gym? The hypothesis tested is;
H0: The proportion of females for unisex gym is not different from 50%
HA: The proportion of females for unisex gym is not different from 50%
The second hypothesis almost tested the same thing as the first only that in this case the males were involved
instead of the females. The hypothesis tested is;
H0: The proportion of males for unisex gym is not different from 50%
HA: The proportion of males for unisex gym is not different from 50%
The third hypothesis compared the proportions for the males and the females. That is, it sought to test whether the
proportions of the males is the same as that of the females. The hypothesis tested is;
H0: The proportion of males for unisex gym is not different from the proportion of females for unisex
HA: The proportion of males for unisex gym is different from the proportion of females for unisex
The last hypothesis did not test on proportions but rather the average. It sought to understand whether there is a
significant difference in the average time spend on Cardio by the females and the males. The hypothesis tested is;
H0: The average time spent on cardio by the males is not different from that of the females
HA: The average time spent on cardio by the males is different from that of the females

The p-value is reported for the hypothesis tests since it is the value that is used to compare with the significance
level. If the p-value is less than the significance level then the null hypothesis is rejected.
It is much easier to sue a computer to perform hypothesis tests rather than using hand.

References
Boix, C., & Frances , R. (2004). Bones of Contention: The Political Economy of Height Inequality.
Clark, A. (2008). A Farewell to Alms: A Brief Economic History of the World.

Appendix
I. Empirical View
Human height is determined by a combination of genetics and environmental factors making it an active
area of research in both the sciences and social sciences. Recent breakthroughs in sequencing the human
genome have allowed identification of 697 genetic variants that influence the height of an
individual.1 Although genetics plays an important role in understanding variation within a given population,
human growth can be limited by poor childhood nutrition and illness. This makes height strongly correlated
with living standards and hence a good proxy for them. Changes to heights over time and within countries
paints a picture of economic development. One major advantage of using height as a proxy is the
availability of data in the pre-statistical period.
It is important to stress that height is not used as a direct measure of well-being. In the absence of any
abnormality or extremes, we should not expect that changing an individual’s height makes them any more
or less happy all other things being equal.
I.1 Global Perspective of Increase of Human Height
Human height has steadily increased over the past two centuries across the globe. This trend is in line with
general improvements in health and nutrition during this period. Historical data on heights tends to come
from soldiers (conscripts), convicted criminals, slaves and servants. It is for this reason much of the
historical data focuses on men. Recent data on heights uses additional sources including surveys and
medical records.
Height development by world regions – Baten & Blum (2012)2
I.2 Human Heights in Early Europeans

In the chart below we see human heights in males and females in the Eastern Mediterranean region from
the Upper Paleolithic (before 16,000 BC) through to the 20th century.
I.3 The Last Two Millennia
Over the last two millennia, human height, based off of skeletal remains, has stayed fairly steady,
oscillating around 170cm. With the onset of modernity, we see a massive spike in heights in the developed
world. It is worth noting that using skeletal remains is subject to measurement error with respect to the
estimated height and time period.
Male heights from skeletons in Europe, 1-2000 – Clark3
I.4 Mesolithic Times, Middle Ages, Subsistence Societies and
Modern Foragers
Though in general, human height has increased around the world, the following tables demonstrate that this
trend is not universal. This first table shows the heights of men in modern foraging and subsistence
societies. Many of these heights are lower than in the second table, which shows the heights of skeletal
remains in various pre-modern periods. This would seem to indicate that human height is closely related to
the lifestyle prevalent in a society, including the type of food consumed, the type of work undertaken, and
the climate.
Heights of adult males in modern foraging and subsistence societies – Clark (2008)4
Period Group Location Ages Height (centimeters)
1892 Plains Indians (a) United States 23–49 172
1970s Anbarra (b) Australia Adults 172*
1970s Rembarranga (c) Australia Adults 171*
1910 Alaskan Inuit (d) United States Adults 170*
1890 Northern Pacific Indians (e) United States Adults 167*

Period Group Location Ages Height (centimeters)
1944 Sandawe (f) Tanzania Adults 167*
1891 Shoshona (g) United States 20–59 166
1970s Fox Basin Inuit (c) Canada Adults 166*
1880s Solomon Islanders (h) Solomon Is. Adults 165*
1906 Canadian Inuitd (d) Canada Adults 164*
1969 !Kung (i) Bostwana 21–40 163
1980s Ache (j) Paraguay Adults 163*
1970s Hadza (c) Tanzania Adults 163*
1985 Hiwi (j) Venezuela Adults 156*
1980s Batak (c) Philippines Adults 155*
1980s Agta (c) Philippines Adults 155*
1980s Aka (c) Central African Republic Adults 155*
Heights from skeletal remains by period, from mesolithic times until now, globally – Clark (2008)5
Period Location Observations Height (centimeters)
Mesolithic (a) Europe 82 168
Neolithic (a,b) Europe 190 167
Denmark 103 173
1600–1800 ( c) Holland 143 167
1700–1800 ( c) Norway 1956 165
1700–1850 ( c) London 211 170
Pre-Dynastic (d) Egypt 60 165
Dynastic (d) Egypt 126 166
2500 BC (e) Turkey 72 166
1700 BC (f) Lerna, Greece 42 166
2000–1000 BC (g) Harappa, India — 169
300 BC–AD 250 (h) Japan (Yayoi) 151 161
1200–1600 (h) Japan (medieval) 20 159
1603–1867 (h) Japan (Edo) 36 158
1450 (i) Marianas, Taumako 70 174
1650 (i) Easter Island 14 173

Period Location Observations Height (centimeters)
1500–1750 (i) New Zealand 124 174
1400–1800 (i) Hawaii — 173
I.5 The Effects of Immigration on Height Differences
In a pioneering study of Japanese immigrants to Hawaii published in 1939, Harry Shapiro found there to be
a significant difference between the heights of Hawaiian-born Japanese and the Japanese immigrant
population.6Shapiro concluded that environmental factors, particularly diet and healthcare, play a
significant role in determining height and other physical characteristics. The underlying idea here is that
migration from poor countries to rich ones may lead to dramatic changes between generations. In a similar
study, Marcus Goldstein (1943) found there to be differences in the heights and other characteristics of the
children of Mexican immigrants and their parents, as well as with native born Mexican children.7
II. Correlates, Determinants &
Consequences
II.1 Other Development Indicators and Human Height
Since height is used as a proxy for development, it should be strongly correlated with other indicators of
development. The most common indicators of development are income, education and health.
Correlation between (log) income per capita and height – Baten & Blum (2012)8
There appears to be a very strong positive correlation between income and height.
Wages, education and height of males in Brazil and the United States – Todaro & Smith (2011)9

We see here that wages and height are positively correlated in Brazil, with a much weaker correlation
in the United States. There exists a similar relationship between education and height within both
countries. Differences in the compulsory schooling age between the two countries may go some way
to explaining the weaker relationship found in the United States.
Relative mortality by body height rates among Norwegian men aged 40–59, 1963–1979 – Floud,
Fogel, Harris, Hong (2011)10
We see a decline in the relative mortality of men aged 40-59 as height increases, suggesting that
taller individuals may be healthier than their shorter counterparts.
II.2 Why are Africans so tall?

When we exclude the richest countries from our sample we find that the positive correlation between
income and height becomes negative. This result is due to Africans being significantly taller than their
Asian and Latin American counterparts. Research by Deaton also finds a negative correlation exists for
nutrition, risk of disease and women's education when comparing Africa to the same group of countries.
While this may appear to indicate that the difference could be down to genetic factors, Deaton argues that
this is unlikely to be the case since we do not observe any significant difference between the heights of
Americans of Caucasian descent compared with Americans of African descent. Instead Deaton suggests it
may be due to dietary differences and child mortality. Dietary differences (vegetarian diets in Asia) and the
way each respective population has adapted to poverty could generate the observed results.
Additionally, higher rates of child mortality in Africa cause the smallest and weakest children to die early
before reaching maturity.
Average height and real per-capita income in year of birth – Deaton (2007)11
III. Data Quality & Definition
Accurately measuring the height of an individual is a straightforward task and so we should be confident
that there is relatively little measurement error in the recorded data. This is unlikely to be the case when
measuring the height of skeletons. What is more, the techniques used to date skeletal remains (such as
radio carbon dating) only provide a probabilistic estimate.
Another factor to consider is the potential sample bias from the historical sources. Since the height data
is largely composed of soldiers, criminals, salves and servants, these groups may not be representative of
the wider population. This problem has been highlighted by academics researching human height.12 In fact,
the observed drop in height during the industrial revolution -- usually attributed to the negative health
impacts of industrialisation -- can be explained by the labour market conditions that existed at the time.
They argue that "as economies grew, tight labour markets discouraged military enlistments by the most
productive workers, with those enlisting (and being measured) increasingly over-representing the less
advantaged members of society."
By comparing the heights of soldiers in the US army with countries that enforced conscription we can see
the bias more clearly. In countries that had conscription, the average height of conscripts was increasing
over the period, meanwhile in the US where entry was voluntary, the heights of soldiers was falling

Mean heights of volunteer soldiers in the US and in selected countries with conscription - Vox13
Another issue is that there existed almost no systematic recording of women's heights meaning there is very
little historical data on the heights of women.