To estimateTo estimate the the parameters parameters of the of regression the model simple:

\ linear[ \ regressiontext model:

{Second\ Test Grade} = \beta_[ \text0{ + \Secondbeta Test Grade_1} (\ = \betatext_{0First + Test \ Gradebeta})_ +1 (\text{ \Firstvare Testpsilon Grade_i}) +\ ]

epsilonI_i will perform]

we will perform a a linear regression least analysis using squares regression analysis the given data. I'll. compute Let me the calculate estimates the for \ estimates for \(\beta_0beta\_0 () (interinterceptcept)) and and \ \(\betabeta__11 (slope)). ( Lets melope calculate). it now. numpy asimport numpy as np importimport statsmodels stats.api asmodels sm.api

as sm#

Given Given data data

firstfirst_test_test_scores_gr =ades np.array([50 = np.array, 91,([50 ,55 ,91, 5855,, 7958,, 7079,, 7097,, 5197,, 5151,, 59,51 ,69 59,, 56, 6956,, 56, 41 ,56 ,49 ,41, 58, 58, 49, 5899,, 5887,, 9599, 87, ,97 95, ]) 97]) second_test_gradessecond = np_test_scores.array =([70, np.array ([5870, , 7458,, 65 ,74 ,60 ,65, 60, 48,60 ,73 ,60 ,73 ,48 ,66 ,73 , 73, 66, 6767, 70, , 7370, ,76 , 7273,, 6876,, 7372, , 5368,, 7358,, 5350, 58, 50, 55])

Adding, a55])

constant for the Add intercept in the model

aX = sm.add constant for_constant the(first intercept_test_grades term )

X# = Performing linear sm regression.add _constantmodel(first =_test_scores sm)

.#OLS(second Fit the_test_grades, X). regression model model = smfit.()

OLS# Extract(second_testing_scores the, parameters beta X)._fit0,()

Extract beta the estimated_1 = model.params

parameters #beta_ Rounding the results0 to 4 decimal, places beta_0 = beta round(beta_0, _41) =beta_1 model =.params round.round(beta(_1, 4) 4)

betabeta__00,, beta_ beta1_The