Extra Credit.ipynb

Kernel: SageMath 9.2

import numpy as np
import seaborn as sns
import pandas as pd
import matplotlib.pyplot as plt
import scipy.stats as stats
from math import exp
from math import log

#1

#2

It is best to use a one-tailed p-value for x^2 and |x| because they never have negative values, because they are squared or absolute value, therefore you only need to test for the possibility of a relationship on the right side. It will be skewed right, and therefore you should use a one tailed p-value. On the other hand, you should use a two-sided p-value for relative risk because it takes into account effect size and you should make a confidence interval from the data. You need a two sided p-value to make the confidence interval. Also, this data will be more bell-shaped so it is important to look for the possibility of a relationship on both the left and the right. It will be centered around 1 and can have positive and negative values.

#3
numerator = 8117./(8117 + 620)
denominator = 8167/(8167 + 529)

rr = numerator/denominator 
rr
#The relative risk is very high, almost 1. This is so high because the proportion of those who took Vitamin E and didn't get cancer (numerator) is almost as high as the proportion of those who took the placebo and survived. A relative risk of 1 means that there is no effect. Because our relative risk value is almost one, we can conclude that there is likely no effect caused by Vitamin E in relation to cancer. 

0.989213844189003

#4

#5

Even though correlation does not mean causation, we still care about it because it sometimes helps us to predict the future. It allows us to do so without knowing the underlaying cause of the trend. This is important so we can determine what will happen next even if we don't know why.

#6

While the study shows that there is a very high correlation between professors' salaries and the price of alcohol over the last 25 years, we can not draw a conclusion about causation. Correlation does not mean causation. One possible explanation is that as the value of money decreased over the course of the last 25 years, universities had to raise the salaries of their professors so they can live with a good quality of life and stores had to increase the cost of alcohol to maintain good profit margins. Another explanation could be that the demand for alcohol has gone up in the last 25 years so stores are charging more for it. This is a very plausible answer, as prices and salaries do increase over time. At the same time, professors salaries are also increasing because there is more demand because more people are going to college. This is also plausible, however less likely than the first explanation. Finally, professors have been demanding higher salaries over the course of the last 25 years while alcohol producers have gotten greedy and upped their prices. This could also be true, but is less likely than both the first two explanations.

#7 part a

The independent variable is whether or not the patient received sterile or non sterile treatment. The dependent variable is whether or not the patient died. Both of these variables are categorical.

#7 part b

#7 part c

The null hypothesis is that there is no difference in how many people survive if they are treated with the sterile process or the non-sterile process.

#7 part d

#7 part e

#7 part f- 99% CI for relative risk NOTE: I JUST COPIED THIS FROM GROUPME, I HAVENT DONE ANY OF THE REST OF THIS PROBLEM YET SO THATS WHY IT ISNT RUNNING, I JUST WANTED TO SAVE IT WHILE I REMEMBERED
ster_box= ['S']*34 + ['D']*6
cont_box= ['S']*16 + ['D']*19

results= np.zeros(10000)
for i in range(10000):
    sterile_res= np.random.choice(ster_box, Sterile)
    control_res= np.random.choice(cont_box, Control)
    outcome= np.zeros({2, 2})
    outcome[0, 0]= np.sum(sterile_res == 'D')
    outcome[1, 0]= np.sum(sterile_res == 'S')
    outcome[0, 1]= np.sum(control_res == 'D')
    outcome[1, 1]= np.sum(control_res == 'S')
    results[i]= relRisk(outcome)

results.sort()
lower= log(results[49])
upper= log(results[9949])
CI_lower= exp(log(relobs)*2 - upper)
CI_upper= exp(log(relobs)*2 - lower)
print("The caclulated 99% confidence interval is:", (CI_lower, CI_upper))

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-15-c6a863482639> in <module>
      5 results= np.zeros(Integer(10000))
      6 for i in range(Integer(10000)):
----> 7     sterile_res= np.random.choice(ster_box, Sterile)
      8     control_res= np.random.choice(cont_box, Control)
      9     outcome= np.zeros({Integer(2), Integer(2)})
NameError: name 'Sterile' is not defined

#7 part g

#7 part h

#8 part a

#8 part b

#8 part c

#8 part d

#8 part e

#9- uploading data
nenana= pd.read_csv("nenana-1.txt", sep= '\t')
nenana

	Ice Breakup Day of Year	Years Since 1900	Year
0	119.479	17	1917
1	130.398	18	1918
2	122.606	19	1919
3	131.448	20	1920
4	130.279	21	1921
...	...	...	...
98	113.601	115	2015
99	112.652	116	2016
100	120.500	117	2017
101	121.554	118	2018
102	104.014	119	2019

103 rows × 3 columns

#9 part a- visualize data
sns.swarmplot(data= nenana["Ice Breakup Day of Year"])

<AxesSubplot:>

#9 part b

#9 part c

#9 part d

#9 part e

#9 part f

#9 part g- line of best fit
regr= stats.linregress(nenana["Year"], nenana["Ice Breakup Day of Year"])
slope= regr.slope
intercept= regr.intercept

print(slope)
print(intercept)
print(min(nenana["Year"]))
print(max(nenana["Year"]))

-0.08441029301937356
289.7854275359135
1917
2019

X_plot= np.linspace(1917, 2019, 100)
Y_plot= -0.08441029301937356*X_plot + 289.7854275359135

sns.lmplot(data= nenana, x= "Year", y= "Ice Breakup Day of Year", fit_reg=False)
plt.plot(X_plot, Y_plot)

[<matplotlib.lines.Line2D object at 0x7f0f677b41c0>]

#9 part h- 99% CI for line of best fit
sims= np.zeros(10000)
for i in range(10000):
    newdata = nenana.sample(len(nenana), replace = True)
    reg= stats.linregress(newdata["Year"], newdata["Ice Breakup Day of Year"])
    sims[i]= reg.slope
sims.sort()
lower= 2*slope - sims[9949]
upper= 2*slope - sims[49]
p= sns.displot(sims, kde=False)
plt.axvline(lower, color = "red")
plt.axvline(upper, color = "red")
[lower,upper]

[-0.14038740420832185, -0.02912926026048468]

#9 part i

#10- uploading data
reaction= pd.read_csv("acid-phosphatase-corrected.csv")
reaction

	Temperature	Initial Reaction Rate
0	298.0	0.05
1	303.0	0.07
2	308.0	0.12
3	313.0	0.20
4	313.0	0.18
5	318.0	0.34
6	323.0	0.48
7	328.0	0.79
8	333.0	0.98
9	335.0	1.02
10	333.5	1.04
11	338.0	1.10
12	343.0	0.98
13	298.0	0.04
14	343.7	1.00
15	353.0	0.53
16	353.0	0.58
17	353.0	0.61
18	338.0	1.07
19	348.0	0.74
20	348.0	0.72

#10 part a- visualize data
sns.violinplot(data= reaction["Initial Reaction Rate"])

<AxesSubplot:>

#10 part b

#10 part c

#10 part d

#10 part e

#10 part f

#10 part f- line of best fit
regr= stats.linregress(reaction["Temperature"], reaction["Initial Reaction Rate"])
slope= regr.slope
intercept= regr.intercept

print(slope)
print(intercept)
print(min(reaction["Temperature"]))
print(max(reaction["Temperature"]))

0.015257856886188529
-4.426794241937526
298.0
353.0

#10 part g- line of best fit
X_plot= np.linspace(298, 353, 100)
Y_plot= 0.015257856886188529*X_plot - 4.426794241937526

sns.lmplot(data= reaction, x= "Temperature", y= "Initial Reaction Rate", fit_reg=False)
plt.plot(X_plot, Y_plot)

[<matplotlib.lines.Line2D object at 0x7f0f6762fd30>]

#10 part h- 99% CI for line of best fit
sims= np.zeros(10000)
for i in range(10000):
    newdata = reaction.sample(len(reaction), replace = True)
    reg= stats.linregress(newdata["Temperature"], newdata["Initial Reaction Rate"])
    sims[i]= reg.slope
sims.sort()
lower= 2*slope - sims[9949]
upper= 2*slope - sims[49]
p= sns.displot(sims, kde=False)
plt.axvline(lower, color = "red")
plt.axvline(upper, color = "red")
[lower,upper]

[0.005167793391557696, 0.024583951244403678]

#10 part i