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Q59Section C (Section C)59 of 60

Sum of least squares regression estimates

Consider the linear regression model yi=β0+β1xi+ϵi,i=1,2,…,6y_i=\beta_0+\beta_1 x_i+\epsilon_i, i=1, 2, \dots, 6, where β0\beta_0 and β1\beta_1 are unknown parameters and ϵi\epsilon_i's are independent and identically distributed random variables having N(0,1)N(0, 1) distribution. The data on (xi,yi)(x_i, y_i) are given in the following table.

| xix_i | 1.0 | 2.0 | 2.5 | 3.0 | 3.5 | 4.5 | |—|—|—|—|—|—|—| | yiy_i | 2.0 | 3.0 | 3.5 | 4.2 | 5.0 | 5.4 |

If β^0\hat{\beta}_0 and β^1\hat{\beta}_1 are the least squares estimates of β0\beta_0 and β1\beta_1, respectively, based on the above data, then β^0+β^1\hat{\beta}_0+\hat{\beta}_1 equals __________ (round off to 22 decimal places)

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