A simple linear regression model is used to relate a response variable Y and a predictor X.

The relation is Y= 9 + 9X + where V()=7.05.*

A sample was collected. The data, in the order it was collected, is as follows:

( 1.6 , 2 ) ( 1.3 , 2.7 ) ( 3 , 3 ) ( 2.7 , 2 ) ( 2 , 3 ) ... ... ... ...

Determine the fitted value when X=1.3, b0=4, and b1=2.

c003.p0310.q002

Question

A simple linear regression model is used to relate a response variable Y and a predictor X.

The relation is Y= 9  + 9X +  where V()=7.05.*

A sample was collected. The data, in the order it was collected, is as follows:

( 1.6 , 2 )	( 1.3 , 2.7 )	( 3 , 3 )	( 2.7 , 2 )	( 2 , 3 )	... ... ... ...

Determine the fitted value when X=1.3, b0=4, and b1=2.

c003.p0310.q002

BlackTom AI · Accepted Answer

The fitted value (predicted response) from a simple linear regression uses the formula ŷ = b0 + b1·x, so plug in the given estimates directly.

1) Relevant concepts:
- Simple linear regression predicts the response using ŷ = b0 + b1 x, where b0 is the intercept and b1 the slope.
- The fitted value ŷ at a specific X uses the estimated coefficients b0 and b1; the sample data or true model parameters are not needed once b0 and b1 are given.

2) Step-by-step reasoning and calculation:
- Given b0 = 4 and b1 = 2, and we want the fitted value at X = 1.3.
- Use the prediction formula: ŷ = b0 + b1·X.

3) Intermediate arithmetic:
- Compute b1·X = 2 · 1.3 = 2.6.
- Add the intercept: ŷ = 4 + 2.6 = 6.6.

4) Why this is correct:
- The definition of the fitted (predicted) value in linear regression is exactly this linear combination of the intercept and slope times X. Using the provided estimated coefficients yields the predicted response 6.6 for X = 1.3.

5) Common incorrect approaches (why they would be wrong):
- Using the original model coefficients shown earlier in the prompt (for example 9 and 9) would give 9 + 9·1.3 = 20.7, which is incorrect here because the question explicitly asks to use b0 = 4 and b1 = 2.
- Forgetting to multiply b1 by X or misplacing the intercept would also produce wrong results.

6) Conclusion:
- The fitted (predicted) value at X = 1.3 with b0 = 4 and b1 = 2 is 6.6, which matches the expected answer.

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