1. Create a data file with the name results.txt that contains a copy of the two columns under the titles xn and tn in the table below. (Do not copy these titles and the bottom row of the table to the file.) Then add some code to the file linearRegression.py to call the two functions there to complete the following tasks:

   First fit a linear function to the data stored in file olympic_women.txt, and then print the coefficients to python IDLE shell. (Use print(*). Then plot the data as dots and the linear function you obtained in problem a as a line. Then predict the womens winning time at the 2012 and 2016 Olympic games and print them to python IDLE shell.

import numpy as np

import matplotlib.pyplot as plt

#This function fits a linear function to the data. The data

#is stored in matrix variable f where the 1st column is

#attribute values and the 2nd column is target values. The

#function returns the co-efficients

def linear(f):

s = np.size(f[:,0]) #number of sample observations

x = f[:,0] #attribute values: a row vector

t = f[:,1] #target values: a row vector

u = np.ones((1,s)) #a row vector of all 1s

u = np.vstack((u,x))#the transpose of the data matrix shown in the class

v = np.linalg.inv(u@u.T) #v = (uu')^-1

w = v@u@t #w = vut

return(w)

#This function plots the linear function with coefficient vector w

#against the attribue values. It also plots the data as the dots.

#The data is stored in matrix variable f where the 1st column is

#attribute values and the 2nd column is target values.

def plotit(w, f):

s = np.size(f[:,0])#These five lines are the same as in linear(f)

x = f[:,0]

t = f[:,1]

u = np.ones((1,s))

u = np.vstack((u,x))

t1 = w@u

plt.plot(x,t,'o', x,t1)

plt.xlabel('years')

plt.ylabel('winning times')

plt.xlim(1920,2020)

plt.show()

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