Business Statistics: Analyzing Customer Preferences for Unisex Gyms

This report analyzes customer preferences regarding a unisex gym, focusing on data from a survey of 100 customers. It uses statistical methods to summarize data, calculate confidence intervals, and perform hypothesis tests. The analysis covers variables such as gender, preference for a unisex gym, time spent on cardio machines, and time spent on weight machines. Key findings include a higher proportion of female respondents preferring a unisex gym compared to males, and differences in the average time spent on cardio machines between genders. The report also includes calculations of confidence intervals for the proportion of males and females preferring a unisex gym, as well as test statistics to check if the proportion of each group wanting a unisex gym is above 50%. The analysis uses webpages to perform hypothesis tests and concludes by discussing the structure of a good statistical report and explaining the hypothesis tests used.

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