Statistics Final

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Question:

Due to financial hardship, the Nyke shoe company feels they only need to make one size of shoes, regardless of gender or height. They have collected data on gender, shoe size, and height and have asked you to tell them if they can change their business model to include only one size of shoes – regardless of height or gender of the wearer. In no more 5-10 pages (including figures), explain your recommendations, using statistical evidence to support your findings. The data found are below:

Shoe Size| Height| Gender|
5.00| 63.00| Female|
7.50| 70.00| Female|
9.00| 70.00| Female|
7.00| 64.00| Male|
11.00| 72.00| Male|
12.00| 72.00| Male|
14.00| 76.00| Male|
7.00| 66.00| Female|
7.50| 71.00| Female|
8.00| 68.00| Female|
10.50| 71.00| Male|
11.00| 71.00| Male|
6.50| 65.00| Female|
7.00| 67.00| Female|
7.50| 70.00| Female|
10.00| 69.00| Male|
12.00| 69.00| Male|
6.50| 65.00| Female|
10.50| 72.00| Male|
12.00| 73.00| Male|
6.00| 60.00| Female|
6.50| 64.00| Female|
10.00| 72.00| Female|
9.50| 69.00| Male|
11.50| 70.00| Male|
14.00| 75.00| Male|
6.50| 63.00| Female|
13.50| 77.00| Male|
7.00| 68.00| Female|
9.50| 68.00| Male|
13.00| 72.00| Male|
11.00| 73.00| Male|
6.00| 62.00| Female|
7.00| 66.00| Female|
7.50| 70.00| Female|

To start out, let’s examine if there is a correlation between the shoe size and the height. Using Excel, we obtain the following table (see sheet CORRELATION)

 | Shoe size| Height|
Shoe size| 1|  |
Height| 0.86434| 1|

As we can see that the correlation coefficient R=0.86434 is positive and close to 1, it suggests that there is a strong positive relationship between the shoe size and the height.

Next, let’s formulate the following data (see SHOW SIZE DATA)

Using Excel to get the following descriptive statistics for both variables: shoe sizes for females (FEMALE) and shoe sizes for males (MALES) we get the below charts:

(FEMALE)
Descriptive Statistics |
 |  |
Mean| 7.111111111|
Standard Error| 0.266775577|
Median| 7|
Mode| 7.5|
Standard Deviation| 1.131832917|
Sample Variance| 1.281045752|
Kurtosis| 1.83070498|
Skewness| 0.854222139|
Range| 5|
Minimum| 5|
Maximum| 10|
Sum| 128|
Count| 18|
(MALE)
Descriptive Statistics |
 |  |
Mean| 11.29411765|
Standard Error| 0.437360952|
Median| 11|
Mode| 11|
Standard Deviation| 1.8032854|
Sample Variance| 3.251838235|
Kurtosis| 0.687533151|
Skewness| -0.454073545|
Range| 7|
Minimum| 7|
Maximum| 14|
Sum| 192|
Count| 17|

From the above tables, we can see that the average shoe size:

For females was 7.11 with a standard deviation of 1.13

For males was 11.29 with a standard deviation of 3.25.

It seems that there is a significant difference between the average shoe sizes for females and for males.

To find out a more correct answer, let’s conduct an independent sample t-test in the following.

Denote by the average shoe sizes for males and females, respectively.

Here are the two hypotheses:
Null hypothesis H0:
Alternative hypothesis Ha:

We conduct an independent sample t-test using Excel, and obtain the following output (see sheet T-TEST)

t-Test: Two-Sample Assuming Unequal Variances|
| | |
 | shoe size(female)| shoe size (male)|
Mean| 7.111111111| 11.29411765|
Variance| 1.281045752| 3.251838235|
Observations| 18| 17|
Hypothesized Mean Difference| 0|  |
df| 27|  |
t Stat| -8.165111165|  |
P(T<=t) one-tail| 4.52926E-09|  |
t Critical one-tail| 1.703288423|  |
P(T<=t) two-tail| 9.05852E-09|  |
t Critical two-tail| 2.051830493|  |

From the above output, we can see that the p-value is 9.05852E-09, which is smaller than 0.05 (if we select a 0.05...
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