Question

fiona wrote the predicted and residual values for a data set using the line of best fit y=3.71x-8.85.

x	given	predicted	residual
2	-1.3	-1.43	-0.13
3	1.9	2.28	-0.38
4	6.2	5.99	0.21

which statements are true about the table? select three options.

• the data point for x = 1 is above the line of best fit.
• the residual value for x = 3 should be a positive number because the data point is above the line of best fit.
• fiona made a subtraction error when she computed the residual value for x = 4.
• the residual value for x = 2 should be a positive number because the given point is above the line of best fit.
• the residual value for x = 3 is negative because the given point is below the line of best fit.

Explanation:

Step1: Analyze x=1

Residual = Given - Predicted = -5.1 - (-5.14) = 0.04. Since residual > 0, data point is above the line. So first statement is true.

Step2: Analyze x=3

Given = 1.9, Predicted = 2.28. Residual = 1.9 - 2.28 = -0.38 (negative). So second statement (residual positive) is false.

Step3: Analyze x=4

Residual = 6.2 - 5.99 = 0.21. Let's check: 6.2 - 5.99 = 0.21, correct? Wait, 5.99 + 0.21 = 6.2, yes. Wait, no error? Wait, no, wait: Wait, the line is y=3.71x -8.85. For x=4, predicted y=3.71*4 -8.85=14.84 -8.85=5.99. Given is 6.2. Residual=6.2 -5.99=0.21, correct. Wait, but the third statement says Fiona made a subtraction error. Wait, maybe I miscalculated? Wait, 6.2 -5.99: 6.2 - 5.99 = 0.21, that's correct. Wait, no, maybe the problem is different? Wait, no, the third statement is "Fiona made a subtraction error when she computed the residual value for x = 4." But according to calculation, residual is 0.21, which is correct. Wait, maybe I made a mistake? Wait, 3.71*4=14.84, 14.84 -8.85=5.99. Given is 6.2. 6.2 -5.99=0.21. So residual is correct. So third statement is false? Wait, no, maybe I messed up. Wait, the fourth statement: x=2, Given=-1.3, Predicted=-1.43. Residual=-1.3 - (-1.43)=0.13? Wait, no, the table says residual is -0.13. Wait, wait, residual is Given - Predicted. So -1.3 - (-1.43)= -1.3 +1.43=0.13. But the table has -0.13. Oh! Wait, here's the error. Wait, for x=2, residual should be -1.3 - (-1.43)=0.13, but table says -0.13. Wait, no, the question's third statement is about x=4. Wait, no, let's re-examine. Wait, the third statement: "Fiona made a subtraction error when she computed the residual value for x = 4." Wait, for x=4, residual is 6.2 -5.99=0.21. But let's check the table: the table has residual 0.21, which is correct. Wait, maybe the fourth statement: x=2, Given=-1.3, Predicted=-1.43. Residual should be -1.3 - (-1.43)=0.13, but table has -0.13. So Fiona made an error at x=2, but the third statement is about x=4. Wait, maybe I misread. Wait, the third statement: "Fiona made a subtraction error when she computed the residual value for x = 4." Let's recalculate x=4: 6.2 -5.99=0.21, which is correct. So third statement is false. Wait, the fourth statement: "The residual value for x = 2 should be a positive number because the given point is above the line of best fit." Given x=2, given y=-1.3, predicted y=-1.43. Since -1.3 > -1.43 (given > predicted), residual should be positive (given - predicted = -1.3 - (-1.43)=0.13), but table has -0.13. So Fiona made a mistake here, but the fourth statement says "should be positive", which is true. Wait, the fourth statement: "The residual value for x = 2 should be a positive number because the given point is above the line of best fit." Given is -1.3, predicted is -1.43. -1.3 is greater than -1.43 (since -1.3 is to the right of -1.43 on number line), so data point is above the line. Residual = given - predicted = -1.3 - (-1.43)=0.13 (positive). But table has -0.13, so the statement says "should be positive", which is true. Wait, so fourth statement is true? Wait, no, the fourth statement is "The residual value for x = 2 should be a positive number because the given point is above the line of best fit." So that's true. Wait, but earlier analysis: let's list all statements:

1. The data point for x = 1 is above the line of best fit. (True, residual 0.04 >0)
2. The residual value for x = 3 should be a positive number because the data point is above the line of best fit. (False, residual is -0.38)
3. Fiona made a subtraction error when she computed the residual value for x = 4. (False, residual is 0.21, correct)
4. The residual value for x = 2 should be a positive number because the given point is above the line of best fit. (True, given -1.3 > predicted -1.43, so residual positive)
5. The residual value for x = 3 is negative because the given point is below the line of best fit. (True, given 1.9 < predicted 2.28, so residual negative)

Wait, the question says "Select three options". So the true ones are:

1. The data point for x = 1 is above the line of best fit.

4. The residual value for x = 2 should be a positive number because the given point is above the line of best fit.

5. The residual value for x = 3 is negative because the given point is below the line of best fit.

Wait, let's recheck:

For x=1: residual 0.04 (positive) → above line → true.

For x=2: given -1.3, predicted -1.43. -1.3 > -1.43 → data point above line → residual should be positive (given - predicted = 0.13), but table has -0.13. So the statement "should be positive" is true (because the given point is above the line, so residual should be positive, even though Fiona made a mistake in the table? Wait, the statement is "The residual value for x = 2 should be a positive number because the given point is above the line of best fit." So regardless of the table's error, the logic is: if given > predicted, residual is positive. So since -1.3 > -1.43, residual should be positive. So the statement is true.

For x=3: given 1.9, predicted 2.28. 1.9 < 2.28 → data point below line → residual negative (1.9 - 2.28 = -0.38). So the statement "The residual value for x = 3 is negative because the given point is below the line of best fit." is true.

So the three true statements are:

1. The data point for x = 1 is above the line of best fit.

4. The residual value for x = 2 should be a positive number because the given point is above the line of best fit.

5. The residual value for x = 3 is negative because the given point is below the line of best fit.

Wait, but the options are:

Option 1: The data point for x = 1 is above the line of best fit. → True.

Option 2: The residual value for x = 3 should be a positive number because the data point is above the line of best fit. → False (data point is below, residual negative).

Option 3: Fiona made a subtraction error when she computed the residual value for x = 4. → False (residual correct).

Option 4: The residual value for x = 2 should be a positive number because the given point is above the line of best fit. → True (given > predicted, so residual positive).

Option 5: The residual value for x = 3 is negative because the given point is below the line of best fit. → True (given < predicted, residual negative).

So the three true options are 1, 4, 5.

Answer:

1. The data point for \( x = 1 \) is above the line of best fit.
2. The residual value for \( x = 2 \) should be a positive number because the given point is above the line of best fit.
3. The residual value for \( x = 3 \) is negative because the given point is below the line of best fit.