Question:
Consider the linear regression model yi = β0 + β1xi + ∈i , i = 1, 2, … , 6, where β0 and β1 are unknown parameters and ∈i ’s are independent and identically distributed random variables having N(0, 1) distribution. The data on (xi, yi) are given in the following table.xi 1.0 2.0 2.5 3.0 3.5 4.5 yi 2.0 3.0 3.5 4.2 5.0 5.4
If \(\hat{\beta}_0\) and \(\hat{\beta}_1\) are the least squares estimates of β0 and β1, respectively, based on the above data, then \(\hat{β}0 + \hat{β}1\) equals __________ (round off to 2 decimal places)
Consider the linear regression model yi = β0 + β1xi + ∈i , i = 1, 2, … , 6, where β0 and β1 are unknown parameters and ∈i ’s are independent and identically distributed random variables having N(0, 1) distribution. The data on (xi, yi) are given in the following table.
If \(\hat{\beta}_0\) and \(\hat{\beta}_1\) are the least squares estimates of β0 and β1, respectively, based on the above data, then \(\hat{β}0 + \hat{β}1\) equals __________ (round off to 2 decimal places)
| xi | 1.0 | 2.0 | 2.5 | 3.0 | 3.5 | 4.5 |
| yi | 2.0 | 3.0 | 3.5 | 4.2 | 5.0 | 5.4 |
Updated On: Nov 24, 2025
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Correct Answer: 2
Solution and Explanation
1. Organize the Data and Calculate Summations
First, we create a table to calculate the necessary sums: ∑xi, ∑yi, ∑xi2, and ∑xiyi.
| i | xi | yi | xi2 | xiyi |
|---|---|---|---|---|
| 1 | 1.0 | 2.0 | 1.00 | 2.00 |
| 2 | 2.0 | 3.0 | 4.00 | 6.00 |
| 3 | 2.5 | 3.5 | 6.25 | 8.75 |
| 4 | 3.0 | 4.2 | 9.00 | 12.60 |
| 5 | 3.5 | 5.0 | 12.25 | 17.50 |
| 6 | 4.5 | 5.4 | 20.25 | 24.30 |
| Sum | 16.5 | 23.1 | 52.75 | 71.15 |
Number of observations, n=6.
2. Calculate the Means
xˉ=n∑xi=616.5=2.75
yˉ=n∑yi=623.1=3.85
3. Calculate the Slope Estimate (β^1)
The formula for the slope is:
β^1=∑xi2−nxˉ2∑xiyi−nxˉyˉ
Numerator (Sxy):
71.15−6(2.75)(3.85)=71.15−63.525=7.625
Denominator (Sxx):
52.75−6(2.75)2=52.75−6(7.5625)=52.75−45.375=7.375
Slope:
β^1=7.3757.625≈1.0339
4. Calculate the Intercept Estimate (β^0)
The formula for the intercept is:
β^0=yˉ−β^1xˉ
Substitute the values:
β^0=3.85−(1.0339)(2.75)
β^0=3.85−2.8432
β^0≈1.0068
5. Calculate the Final Sum (β^0+β^1)
β^0+β^1=1.0068+1.0339
β^0+β^1=2.0407
Rounding to two decimal places:
Answer: 2.04
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