For this question, considering "multivariate case", we run the code:

# import package

import pandas as pd

import numpy as np

from scipy import stats

# read data

Female$\_$data $=$ pd$.$read$\_$csv$($r$"$C:$\backslash$Users$\backslash$female$.$csv$",$header$=$None$)$

Male$\_$data $=$ pd$.$read$\_$csv$($r$"$c:$\backslash$Users$\backslash$male$.$csv$",$header$=$None$)$

N $=$ len$($Female$\_$data$)$

# log transformation

Female$\_$data$\_$log $=$ np$.$log$1$p$($Female$\_$data$)$

Male$\_$data$\_$log $=$ np$.$log$1$p$($Male$\_$data$)$

# covariance matrices

cov$\_$matrix$\_$female $=$ Female$\_$data$\_$log$.$cov$()$

cov$\_$matrix$\_$male $=$ Male$\_$data$\_$log$.$cov$()$

# number of samples in each group

n$\_$female $=$ len$($Female$\_$data$)$

n$\_$male $=$ len$($Male$\_$data$)$

# pooled covariance matrix

pooled$\_$cov$\_$matrix $=(($n$\_$female $-1)*$ cov$\_$matrix$\_$female $+($n$\_$male $-1)*$ cov$\_$matrix$\_$male$)/($n$\_$female $+$ n$\_$male $-2)$

# mean difference

mean$\_$diff $=$ Female$\_$data$\_$log$.$mean$()-$ Male$\_$data$\_$log$.$mean$()$

# Hotelling's T$-$squared statistic

t$\_$squared $=$ np$.$dot$($mean$\_$diff.T$,$ np$.$dot$($np$.$linalg.inv$($pooled$\_$cov$\_$matrix$),$ mean$\_$diff$))*($n$\_$female $*$ n$\_$male$)/($n$\_$female $+$ n$\_$male$)$

# degrees of freedom

df$\_$t$\_$squared $=$ len$($Female$\_$data.columns$)$

df $=$ n$\_$female $+$ n$\_$male $-2$

# calculate p$-$value after comparison with the F$-$distribution

p$\_$t$\_$squared $=1-$ stats.f$.$cdf$($t$\_$squared, dfn$=$df$\_$t$\_$squared, dfd$=$df$)$

print$("$Hotelling$\text{'}$s T$-$squared $="+$ str$($t$\_$squared$))$

print$("$p $="+$ str$($p$\_$t$\_$squared$))$

# t$-$test for each variable

t$2,$ p$2=$ stats.ttest$\_$ind$($Female$\_$data$\_$log$,$ Male$\_$data$\_$log$)$

print$("$p$($t$)="+$ str$(2*$ p$2))$

for i in range$(0,$ len$($p$))$:

if $2*$ p$[$i$]$ p$2[$i$]$:

print$($Female$\_$data.columns$[$i$]+"$ is significantly different"$)$

else:

print$($Female$\_$data.columns$[$i$]+"$ is not significantly different"$)$

And we get the result:

Hotelling's T$-$squared $=84\mathrm{.}74826838379414$

p $=1\mathrm{.}1102230246251565$e$-16$

p$($t$)=[1\mathrm{.}13631611$e$-044\mathrm{.}24778379$e$-059\mathrm{.}55243601$e$-08]$

which is wrong. Can you help me to correct the code and help me to explain the result? Thank you!:$)$