Linear regression between SAT and GPA

Compare with multilinear regression result：click here

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import statsmodels.api as sm
## beautify the plot made with matplotlib 
import seaborn as sns
sns.set()

In [9]:

data = pd.read_csv('C:\\Users\\Python_practice\\1.01. Simple linear regression.csv')   
##read csv file

In [10]:

data.describe()   ## Pandas method to give the most useful descriptive statistics in the data frame

Out[10]:

	SAT	GPA
count	84.000000	84.000000
mean	1845.273810	3.330238
std	104.530661	0.271617
min	1634.000000	2.400000
25%	1772.000000	3.190000
50%	1846.000000	3.380000
75%	1934.000000	3.502500
max	2050.000000	3.810000

In [11]:

## Predict GPA via SAT
## Reason: SAT is one of the best estimators of intellectual capacity
## Creat scatter plot of "x1 & y"  via  "matplotlib.pyplot"
y = data['GPA']
x1 = data['SAT']
plt.scatter(x1,y)
plt.xlabel('SAT',fontsize=20)
plt.ylabel('GPA',fontsize=20)
plt.show()

In [12]:

## use OLS(最小平方法) to plot with "statsmodels.api"
x = sm.add_constant(x1.to_numpy())  ## Creat regression   y = x0 + b1x1   via  "statsmodels.api", convert x1 to a numpy array
result = sm.OLS(y,x).fit()   ## apply a specific estimation tech(OLS) to obtain the "fit" of the model
result.summary()

Out[12]:

OLS Regression Results
Dep. Variable:	GPA	R-squared:	0.406
Model:	OLS	Adj. R-squared:	0.399
Method:	Least Squares	F-statistic:	56.05
Date:	Fri, 10 Jan 2020	Prob (F-statistic):	7.20e-11
Time:	17:10:15	Log-Likelihood:	12.672
No. Observations:	84	AIC:	-21.34
Df Residuals:	82	BIC:	-16.48
Df Model:	1
Covariance Type:	nonrobust

	coef	std err	t	P>\|t\|	[0.025	0.975]
const	0.2750	0.409	0.673	0.503	-0.538	1.088
x1	0.0017	0.000	7.487	0.000	0.001	0.002

Omnibus:	12.839	Durbin-Watson:	0.950
Prob(Omnibus):	0.002	Jarque-Bera (JB):	16.155
Skew:	-0.722	Prob(JB):	0.000310
Kurtosis:	4.590	Cond. No.	3.29e+04

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 3.29e+04. This might indicate that there are
strong multicollinearity or other numerical problems.

In [13]:

## according to the calculation above, we know b1 = 0.0017 and x0 = 0.275
## use matplotlib.pyplot to draw the regression line
plt.scatter(x1,y)
yhat = 0.0017*x1 + 0.275
fig = plt.plot(x1, yhat, lw=4, c='orange', label='regression line')
plt.xlabel('SAT', fontsize = 20) 
plt.ylabel('GPA', fontsize = 20)
plt.show()