# import libraries

import numpy as np

import matplotlib.pyplot as plt

# NEW!

import scipy.stats as stats

# parameters

n$1=30$ # samples in dataset $1$

n$2=40$ # $...$and $2$

mu$1=1$ # population mean in dataset $1$

mu$2=2$ # population mean in dataset $2$

# generate the data

data$1=$ mu$1+$ np$.$random.randn$($n$1)$

data$2=$ mu$2+$ np$.$random.randn$($n$2)$

# plot them

plt$.$plot$($np$.$zeros$($n$1),$data$1,\text{'}$ro$\text{'},$markerfacecolor$=\text{'}$w$\text{'},$markersize$=14)$

plt$.$plot$($np$.$ones$($n$2),$ data$2,\text{'}$bs$\text{'},$markerfacecolor$=\text{'}$w$\text{'},$markersize$=14)$

plt$.$xlim$([-1,2])$

plt$.$xticks$([0,1],$labels$=[\text{'}$Group $1\text{'},$'Group $2\text{'}])$

plt$.$show$()$

# t$-$test via stats package

# $\_$ind $=$ independent samples

t$,$p $=$ stats.ttest$\_$ind$($data$1,$data$2)$

print$($t$)$

print$($p$)$

# common way to show t$-$test results in a plot

fig $=$ plt$.$figure$($figsize$=(10,4))$

plt$.$rcParams.update$(\{\text{'}$font$.$size':$12\})$ # change the font size

plt$.$plot$(0+$np$.$random.randn$($n$1)/15,$ data$1,\text{'}$ro$\text{'},$markerfacecolor$=\text{'}$w$\text{'},$markersize$=14)$

plt$.$plot$(1+$np$.$random.randn$($n$2)/15,$ data$2,\text{'}$bs$\text{'},$markerfacecolor$=\text{'}$w$\text{'},$markersize$=14)$

plt$.$xlim$([-1,2])$

plt$.$xticks$([0,1],$labels$=[\text{'}$Group $1\text{'},$'Group $2\text{'}])$

# set the title to include the t$-$value and p$-$value

plt$.$title$($f$\text{'}$t $=\{$t:$\mathrm{.}2$f$\},$ p$=\{$p:$\mathrm{.}3$f$\}\text{'})$

plt$.$show$()$

Run the code and observe its functionality.

Provide an explanation of why the T$-$Test is used in the statistical analysis.

Discuss potential scenarios and use cases where the T$-$Test is relevant and valuable in machine learning and data science.

Share your personal insights on when you have previously used the T$-$Test in your data science or machine learning projects $($if applicable$)$ and describe the specific use cases.

This assignment offers an opportunity to deepen your understanding of statistical analysis in the context of deep learning and to apply this knowledge to real$-$world scenarios.