Question

a regression is performed on the table below.

$x$	$sqrt{x}$	$y$
$5.9$	$2.43$	$27.2$
$8.1$	$2.85$	$31.0$
$10.8$	$3.29$	$34.3$
$14.5$	$3.81$	$38.8$

which of the following functions best fits the data?
a. $\bigcirc$ $y = 6.5sqrt{x}+7.2$
b. $\bigcirc$ $y = 7.9sqrt{x}+8.5$
c. $\bigcirc$ $y = 7.2sqrt{x}+6.5$
d. $\bigcirc$ $y = 8.5sqrt{x}+7.9$

Explanation:

Step1: Test Option A

Substitute \(\sqrt{x} = 2.02\) into \(y = 6.5\sqrt{x}+7.2\):
\(y = 6.5\times2.02 + 7.2 = 13.13 + 7.2 = 20.33\).
The actual \(y\) is \(24.9\), difference is \(24.9 - 20.33 = 4.57\).

Step2: Test Option B

Substitute \(\sqrt{x} = 2.02\) into \(y = 7.9\sqrt{x}+8.5\):
\(y = 7.9\times2.02 + 8.5 = 15.958 + 8.5 = 24.458\).
Difference: \(24.9 - 24.458 = 0.442\).

Step3: Test Option C

Substitute \(\sqrt{x} = 2.02\) into \(y = 7.2\sqrt{x}+6.5\):
\(y = 7.2\times2.02 + 6.5 = 14.544 + 6.5 = 21.044\).
Difference: \(24.9 - 21.044 = 3.856\).

Step4: Test Option D

Substitute \(\sqrt{x} = 2.02\) into \(y = 8.5\sqrt{x}+7.9\):
\(y = 8.5\times2.02 + 7.9 = 17.17 + 7.9 = 25.07\).
Difference: \(25.07 - 24.9 = 0.17\).

Check another point (e.g., \(\sqrt{x} = 3.81\), \(y = 38.8\)):

• Option D: \(y = 8.5\times3.81 + 7.9 = 32.385 + 7.9 = 40.285\). Difference: \(40.285 - 38.8 = 1.485\).
• Option B: \(y = 7.9\times3.81 + 8.5 = 30.099 + 8.5 = 38.599\). Difference: \(38.8 - 38.599 = 0.201\).

Overall, Option B has smaller differences across points. Wait, recheck Step4: Wait, initial \(\sqrt{x}=2.02\), Option D gave \(25.07\) (close to \(24.9\)), but for \(\sqrt{x}=3.81\), Option B is closer. Wait, maybe miscalculation. Let's recalculate Option B for \(\sqrt{x}=3.81\): \(7.9\times3.81 = 7.9\times(3 + 0.81)=23.7 + 6.399 = 30.099\), \(30.099 + 8.5 = 38.599\), actual \(y=38.8\), difference \(0.201\). Option D: \(8.5\times3.81 = 32.385\), \(32.385 + 7.9 = 40.285\), difference \(1.485\). Now check \(\sqrt{x}=2.43\) (second point, \(y=27.2\)):

• Option B: \(7.9\times2.43 + 8.5 = 19.197 + 8.5 = 27.697\). Difference: \(27.697 - 27.2 = 0.497\).
• Option D: \(8.5\times2.43 + 7.9 = 20.655 + 7.9 = 28.555\). Difference: \(28.555 - 27.2 = 1.355\).

Option B is more consistent. Wait, but initial Step4: Option D for \(\sqrt{x}=2.02\) was \(25.07\) (close to \(24.9\)), but other points favor B. Wait, maybe I made a mistake. Wait, the first point: \(x=4.1\), \(\sqrt{x}=2.02\), \(y=24.9\).

Option B: \(7.9*2.02=15.958 +8.5=24.458\) (diff 0.442)
Option D: \(8.5*2.02=17.17 +7.9=25.07\) (diff 0.17)

Second point: \(x=5.9\), \(\sqrt{x}=2.43\), \(y=27.2\)
Option B: \(7.9*2.43=19.197 +8.5=27.697\) (diff 0.497)
Option D: \(8.5*2.43=20.655 +7.9=28.555\) (diff 1.355)

Third point: \(x=8.1\), \(\sqrt{x}=2.85\), \(y=31.0\)
Option B: \(7.9*2.85=22.515 +8.5=31.015\) (diff 0.015)
Option D: \(8.5*2.85=24.225 +7.9=32.125\) (diff 1.125)

Ah! Here, Option B at \(\sqrt{x}=2.85\) gives \(31.015\), almost \(31.0\). That's very close. So Option B is better. Wait, but earlier I thought D was close, but third point shows B is very close. So the correct answer is B.

Answer:

B. \( y = 7.9\sqrt{x} + 8.5 \)