%matplotlib inline
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
import pandas as pd

1. Covid Data

2.
perform Chi Squared Test, null hypothesis is that both groups come from the same populations (i.e. vaccine has no effect)
Results come back statistically insignificant but we know that that is not true, we need to measure the size of the effect

#3

Code Appendix

#2
data=([8, 162],[18190, 18163])
df1 = pd.DataFrame(data, columns = ['Drug', 'Placebo'], index=['Infected', 'Not Infected'])
df1

#observed chi squared

inf=[8, 162]
notinf= [18190, 18163]
covid=[inf,notinf]
covidarr=np.array(covid)
a=covidarr[0,0]
b=covidarr[0,1]
c=covidarr[1,0]
d=covidarr[1,1]
N=a+b+c+d
DI=((a+b)/N)*(a+c)
PI=((a+b)/N)*(b+d)
DN=((c+d)/N)*(a+c)
PN=((c+d)/N)*(b+d)
ochisquared=(((a-DI)**2)/DI)+(((b-PI)**2)/PI)+(((c-DN)**2)/DN)+(((d-PN)**2)/PN)
ochisquared

#simulated chi squared

result=[]
drug=['d infected']*8+['d not infected']*18190
placebo=['p infected']*162+['p not infected']*18163
covidzeros=np.zeros([2,2])
for i in range(10000):
    drand=np.random.choice(drug, 18198)
    dinfected = drand == ['d infected']
    dnot = drand == ['d not infected']
    covidzeros[0,0]=np.sum(dinfected)
    covidzeros[1,0]=np.sum(dnot)
    prand=np.random.choice(placebo, 18163)
    pinfected = prand == ['p infected']
    pnot = prand == ['p not infected']
    covidzeros[0,1]=np.sum(pinfected)
    covidzeros[1,1]=np.sum(pnot)
    a=covidzeros[0,0]
    b=covidzeros[0,1]
    c=covidzeros[1,0]
    d=covidzeros[1,1]
    N=a+b+c+d
    DI=((a+b)/N)*(a+c)
    PI=((a+b)/N)*(b+d)
    DN=((c+d)/N)*(a+c)
    PN=((c+d)/N)*(b+d)
    chisquared=(((a-DI)**2)/DI)+(((b-PI)**2)/PI)+(((c-DN)**2)/DN)+(((d-PN)**2)/PN)
    result.append(chisquared)

graph=sns.histplot(results)
graph.axvline(ochisquared, color="red")
count=results>=ochisquared
p=np.sum(count=='True')
p/10000