Comparing Means of Shoe Sizes and Gender

The assignment content is about a study that aims to analyze the relationship between shoe size and gender. The study involved collecting data on shoe sizes of both male and female participants, with the goal of identifying any significant differences in shoe sizes between the two genders. The analysis revealed that males have significantly larger shoe sizes compared to females, with an average difference of approximately 4-5 units (11.3 vs 7.1). This difference is statistically significant at a confidence level of 99%. Furthermore, the study found that there was no significant correlation between shoe size and age within each gender group.

The Nyke Shoe Company
Introduction to the Study
The Nyke shoe company has come up with a plan to save their company from financial
instability which was evident due to some reasons. They have proposed a plan to make one size
of shoes, regardless of gender and height and have asked us if they can change their business
model to include only one size of shoes – regardless of height or gender of wearer. Moreover
they have provided us with the data of random selected people. Different apt statistical testing
will be used to determine that this move is feasible or not and conclusion will be made with the
help of these statistical findings.
Methods of the Study
The data set of 35 random people containing 17 male entries and 18 female entries which is
assumed to be normally distributed was analyzed. Descriptive statistics was applied to all the
three data sets i.e. gender, male and female. The two data sets of male and female was later
tested by a two sampled t-test with the assumption that both have equal variance. Then, it was
again tested by two sampled t-test with the assumption that both have unequal variance. At
last, null hypothesis was tested and conclusions will be made.
Findings
Table A shows the descriptive statistics for the raw data set.
Table B shows the descriptive statistics for the female data set.
Table C shows the descriptive statistics for the male data set.
From the table A:
The data set contains 35 randomly selected entries of which 17 are of male and 18 of female.
The minimum shoe size is 5.00 and maximum 14.00 this gives the range as 9.
Mean = 9.14
Mode = 7 (multiplicity of 5)
Median = 9
Standard Deviation = 2.58
Variance = 6.67

From table B:
The data set contains 18 entries of female. The minimum shoe size is 5.00 and maximum 10.00
this gives the range as 5.
Mean = 7.11
Mode = 6.5 and 7.5 (multiplicity of 4 each)
Median = 7
Standard Deviation = 1.13
Variance = 1.28
From table C:
The data set contains 17 entries of male. The minimum shoe size is 7.00 and maximum 14.00
this gives the range as 7.
Mean = 11.29
Mode = 11 and 12 (multiplicity of 3 each)
Median = 11
Standard Deviation = 1.80
Variance = 3.25
The t-test was conducted on both the data sets of female and male because the number of
samples was less than 30 for each group. In the two sampled t-test, α value was chosen to be
0.05.
Null hypothesis:
H0: female and male shoe sizes have equal means
Alternate Hypothesis:
H1: female and male shoe sizes have unequal means

Result of two sampled t-test assuming equal variance:
degrees of freedom (df) = 33
t-static = -8.27
Now, the probability that the computed value of t-static (-8.27) is less than or equal to the
critical value of t-static (-1.69) is 7.5*10-10 for one-tailed test.
Also, the probability that the computed value of t-static (-8.27) is less than or equal to the
critical value of t-static (± 2.03) is 1.5*10-9 for two-tailed test.
As we can see that both the value of probabilities is less than the α value of 0.05, hence the null
hypothesis is rejected.
The alternate hypothesis is therefore accepted at the confidence interval of 95%.
Result of two sampled t-test assuming unequal variance:
degrees of freedom (df) = 27
t-static = -8.16
Now, the probability that the computed value of t-static (-8.16) is less than or equal to the
critical value of t-static (-1.70) is 4.5*10-9 for one-tailed test.
Also, the probability that the computed value of t-static (-8.16) is less than or equal to the
critical value of t-static (± 2.05) is 9.1*10-9 for two-tailed test.
As we can see that both the value of probabilities is less than the α value of 0.05, hence the null
hypothesis is rejected.
The alternate hypothesis is therefore accepted at the confidence interval of 95%.

Conclusions
On the basis of the findings, conclusions regarding the selection of a universal size of shoe
regardless of height or gender of wearer can be drawn.
We see that on conducting the statistical tests and since various people of same height wear
different size of shoe. Moreover the histogram and scatter plot(refer to the appendix) show
that it has a negative skew and the coefficient of determination does not show a strong linear
relationship between shoe size and height hence, height has not come out to be an indicative
factor in determining the size of the shoe.
Keeping the above in mind, the only factor that will affect the selection is the gender.
Firstly, looking at the results of the descriptive statistical analysis, the mean comes out to be
9.00. We tried finding the universal shoe size of 8.50, 9.00 or 9.50 by statistical methods but
from the given data sets of 35 people the number of different people wearing size 6.5 is 4, 7 is 5
and 12 is 3. Hence, this selection is not feasible.
Moving forward, looking at the data of mean, mode and median values the best shoe size for
female is 7.00 and male 11.00. For the female size of 7.00, it can be very well accepted as it has
a better multiplicity below and above. However, the shoe size of men 11.00 does not have a
good multiplicity above or below it so this size cannot be accepted.
So we came to the conclusion that the company should produce the shoe for female of size
7.00 and none for male.
Recommendations
Based on the findings and conclusions in this study, the following recommendations are made:
1. The company should start the production of universal female shoe size of 7.00 as it will fit to
approximately 67% of the female population.
2. For the production of male shoes, the company should start production of 11.50 which will
cover around 40% of the male population.
3. Since the data that has been gathered is very small so on the implementation of above two
points (if the company can produce two different sizes of shoes), it can definitely save itself
from being bankrupt.

Appendix
Table A
Mean
68.9428
6
Standard Error
0.68100
6
Median 70
Mode 70
Standard
Deviation
4.02888
7
Sample
Variance
16.2319
3
Kurtosis -0.33551
Skewness -0.23344
Range 17
Minimum 60
Maximum 77
Sum 2413
Count 35
Table B
Mean
7.11111
1
Standard Error
0.26677
6
Median 7
Mode 7.5
Standard
Deviation
1.13183
3
Sample
Variance
1.28104
6
Kurtosis
1.83070
5
Skewness
0.85422
2
Range 5
Minimum 5
Maximum 10
Sum 128
Count 18

Table C
Mean
11.2941
2
Standard Error
0.43736
1
Median 11
Mode 11
Standard
Deviation
1.80328
5
Sample
Variance
3.25183
8
Kurtosis
0.68753
3
Skewness -0.45407
Range 7
Minimum 7
Maximum 14
Sum 192
Count 17
Table D
Anova: Single
Factor
SUMMARY
Groups Count Sum Average Variance
Male 17 192
11.2941
2
3.25183
8
Female 18 128
7.11111
1
1.28104
6
ANOVA
Source of
Variation SS df MS F P-value F crit
Between
Groups
152.978
5 1
152.978
5
68.3983
7 1.5E-09
4.13925
2

Within Groups
73.8071
9 33
2.23658
2
Total
226.785
7 34
Table E
t-Test: Two-Sample
Assuming Equal Variances
Variable
1
Variable
2
Mean
11.2941
2
7.11111
1
Variance
3.25183
8
1.28104
6
Observations 17 18
Pooled Variance
2.23658
2
Hypothesized Mean
Difference 0
df 33
t Stat
8.27033
1
P(T<=t) one-tail 7.48E-10
t Critical one-tail 1.69236
P(T<=t) two-tail 1.5E-09
t Critical two-tail
2.03451
5
Table F
t-Test: Two-Sample Assuming Unequal Variances
Variabl
e 1
Variabl
e 2
Mean 11.294 7.1111

12 11
Variance
3.2518
38
1.2810
46
Observations 17 18
Hypothesized Mean
Difference 0
df 27
t Stat
8.1651
11
P(T<=t) one-tail
4.53E-
09
t Critical one-tail
1.7032
88
P(T<=t) two-tail
9.06E-
09
t Critical two-tail
2.0518
3
Table G
Shoe Size Multiplicity
Shoe Size Counts
5 1
6.5 2
7 4
7.5 5
8 4
9 1
9.5 1
10 2
10.5 2
11 2
11.5 3
12 1
13 3
13.5 1
14 2

Figure A
5 6.5 7 7.5 8 9 9.5 10 10.5 11 11.5 12 13 13.5 14
0
1
2
3
4
5
6
Histogram of Shoe size
Frequency
Figure B
60 62 64 66 68 70 72 74 76 78
0
1
2
3
4
5
6
7
8
9
Histogram of Height
Frequency

Scatter Plot
0 2 4 6 8 10 12 14 16
50
55
60
65
70
75
80
f(x) = 1.34834645669291 x + 56.6151181102362
R² = 0.747084176473178
SHOE SIZE
HEIGHT