Question

1.
a. write the equation of the graph. y=mx+b
b. what is the coefficient correlation (the value of ( r ))?
c. solve for ( f(250) )

calories per beef hot dog	milligrams of sodium per beef hot dog
181	477
176	425
149	322
184	482
190	587
158	370
139	322

2.
a. write the equation of the graph. ( y=mx+b )
b. what is the coefficient correlation (the value of ( r ))?
c. solve for ( f(120) )

number of minutes	10	85	20	60	35	45	25	70	40	65

Answer:

Problem 1

Part A: Write the equation of the graph (\(y = mx + b\))

To find the linear regression equation \(y = mx + b\) (where \(x\) is calories per beef hot dog and \(y\) is milligrams of sodium per beef hot dog), we can use the formula for the slope \(m\) and the y-intercept \(b\).

First, we calculate the necessary sums:

• \(n = 8\) (number of data points)
• \(\sum x = 186 + 181 + 176 + 149 + 184 + 190 + 158 + 139 = 1363\)
• \(\sum y = 495 + 477 + 425 + 322 + 482 + 587 + 370 + 322 = 3480\)
• \(\sum xy = (186 \times 495) + (181 \times 477) + (176 \times 425) + (149 \times 322) + (184 \times 482) + (190 \times 587) + (158 \times 370) + (139 \times 322)\)
• \(186 \times 495 = 92070\)
• \(181 \times 477 = 86337\)
• \(176 \times 425 = 74800\)
• \(149 \times 322 = 47978\)
• \(184 \times 482 = 88688\)
• \(190 \times 587 = 111530\)
• \(158 \times 370 = 58460\)
• \(139 \times 322 = 44758\)
• \(\sum xy = 92070 + 86337 + 74800 + 47978 + 88688 + 111530 + 58460 + 44758 = 594621\)
• \(\sum x^2 = 186^2 + 181^2 + 176^2 + 149^2 + 184^2 + 190^2 + 158^2 + 139^2\)
• \(186^2 = 34596\)
• \(181^2 = 32761\)
• \(176^2 = 30976\)
• \(149^2 = 22201\)
• \(184^2 = 33856\)
• \(190^2 = 36100\)
• \(158^2 = 24964\)
• \(139^2 = 19321\)
• \(\sum x^2 = 34596 + 32761 + 30976 + 22201 + 33856 + 36100 + 24964 + 19321 = 234775\)

The slope \(m\) is given by:
\[
m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}
\]
Substituting the values:
\[
m = \frac{8 \times 594621 - 1363 \times 3480}{8 \times 234775 - (1363)^2}
\]
First, calculate the numerator:
\(8 \times 594621 = 4756968\)
\(1363 \times 3480 = 4743240\)
Numerator: \(4756968 - 4743240 = 13728\)

Denominator:
\(8 \times 234775 = 1878200\)
\(1363^2 = 1857769\)
Denominator: \(1878200 - 1857769 = 20431\)

So, \(m = \frac{13728}{20431} \approx 0.672\) (Wait, the initial handwritten equation was \(y = 4.00x - 222.00\), maybe I made a mistake in calculation. Let's recalculate the sums correctly.

Wait, maybe the x and y are reversed? Let's check the data again. The table is "Calories per Beef Hot Dog" (x) and "Milligrams of Sodium per Beef Hot Dog" (y). Wait, maybe the initial handwritten equation is correct, so perhaps I mixed up x and y. Let's assume x is sodium and y is calories? No, the table is calories (x) and sodium (y). Wait, maybe the initial equation is a typo, but let's use the correct method.

Alternatively, maybe the problem expects using a calculator or software for linear regression. Let's use the formula for \(b\):
\[
b = \frac{\sum y - m \sum x}{n}
\]

If we take the initial handwritten equation \(y = 4.00x - 222.00\), let's check with x=186: \(y = 4*186 - 222 = 744 - 222 = 522\), but the actual y is 495. Not close. Maybe x is sodium and y is calories? Let's try x as sodium (y) and y as calories (x). Then x (sodium) values: 495, 477, 425, 322, 482, 587, 370, 322. y (calories): 186, 181, 176, 149, 184, 190, 158, 139.

Then \(\sum x = 495 + 477 + 425 + 322 + 482 + 587 + 370 + 322 = 3480\)
\(\sum y = 1363\)
\(\sum xy = 495*186 + 477*181 + 425*176 + 322*149 + 482*184 + 587*190 + 370*158 + 322*139 = 594621\) (same as before)
\(\sum x^2 = 495^2 + 477^2 + 425^2 + 322^2 + 482^2 + 587^2 + 370^2 + 322^2\)

• \(495^2 = 245025\)
• \(477^2 = 227529\)
• \(425^2 = 180625\)
• \(322^2 = 103684\) (twice)
• \(482^2 = 232324\)
• \(587^2 = 344569\)
• \(370^2 = 136900\)
• \(\sum x^2 = 245025 + 227529 + 180625 + 103684 + 232324 + 344569 + 136900 + 103684 = 1574340\)

\(\sum y^2 = 186^2 + 181^2 + 176^2 + 149^2 + 184^2 + 190^2 + 158^2 + 139^2 = 234775\) (same as before)

Now, \(m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}\)
\(n = 8\)
Numerator: \(8*594621 - 3480*1363 = 4756968 - 4743240 = 13728\)
Denominator: \(8*1574340 - 3480^2 = 12594720 - 12110400 = 484320\)
\(m = \frac{13728}{484320} = 0.0283\) (No, that's not right. Clearly, I made a mistake in identifying x and y. Let's use a calculator for linear regression.

Using a calculator, inputting the data:

Calories (x): 186, 181, 176, 149, 184, 190, 158, 139

Sodium (y): 495, 477, 425, 322, 482, 587, 370, 322

Using linear regression:

\(\bar{x} = \frac{1363}{8} \approx 170.375\)

\(\bar{y} = \frac{3480}{8} = 435\)

\(SS_{xx} = \sum (x - \bar{x})^2 = \sum x^2 - \frac{(\sum x)^2}{n} = 234775 - \frac{1363^2}{8} = 234775 - \frac{1857769}{8} = 234775 - 232221.125 = 2553.875\)

\(SS_{xy} = \sum (x - \bar{x})(y - \bar{y}) = \sum xy - \frac{\sum x \sum y}{n} = 594621 - \frac{1363*3480}{8} = 594621 - \frac{4743240}{8} = 594621 - 592905 = 1716\)

Then \(m = \frac{SS_{xy}}{SS_{xx}} = \frac{1716}{2553.875} \approx 0.672\)

\(b = \bar{y} - m\bar{x} = 435 - 0.672*170.375 \approx 435 - 114.49 = 320.51\)

So the equation is \(y = 0.672x + 320.51\), which is different from the handwritten one. Maybe the initial handwritten equation is incorrect, but since the problem has a handwritten equation, maybe we should use that for part C.

Part B: Coefficient of correlation (r)

The formula for \(r\) is:
\[
r = \frac{SS_{xy}}{\sqrt{SS_{xx} SS_{yy}}}
\]
\(SS_{yy} = \sum (y - \bar{y})^2 = \sum y^2 - \frac{(\sum y)^2}{n} = 1574340 - \frac{3480^2}{8} = 1574340 - \frac{12110400}{8} = 1574340 - 1513800 = 60540\)

Then \(r = \frac{1716}{\sqrt{2553.875 * 60540}} = \frac{1716}{\sqrt{154611435}} = \frac{1716}{12434.3} \approx 0.138\) (No, that can't be right. Wait, \(SS_{xx}\) was miscalculated. Wait, \(\sum x^2\) for x (calories) is 186² + 181² + 176² + 149² + 184² + 190² + 158² + 139²:

186² = 34596

181² = 32761

176² = 30976

149² = 22201

184² = 33856

190² = 36100

158² = 24964

139² = 19321

Sum: 34596 + 32761 = 67357; +30976 = 98333; +22201 = 120534; +33856 = 154390; +36100 = 190490; +24964 = 215454; +19321 = 234775. Correct.

\((\sum x)^2 = 1363² = 1857769\), so \(SS_{xx} = 234775 - 1857769/8 = 234775 - 232221.125 = 2553.875\). Correct.

\(\sum y^2\) (sodium) is 495² + 477² + 425² + 322² + 482² + 587² + 370² + 322²:

495² = 245025

477² = 227529

425² = 180625

322² = 103684 (twice)

482² = 232324

587² = 344569

370² = 136900

Sum: 245025 + 227529 = 472554; +180625 = 653179; +103684 = 756863; +103684 = 860547; +232324 = 1,092,871; +344,569 = 1,437,440; +136,900 = 1,574,340. Correct.

\(SS_{yy} = 1,574,340 - (3480)²/8 = 1,574,340 - 12,110,400/8 = 1,574,340 - 1,513,800 = 60,540\). Correct.

\(SS_{xy} = 1,716\) (as before)

Then \(r = \frac{1716}{\sqrt{2553.875 * 60540}} = \frac{1716}{\sqrt{154,611,435}} = \frac{1716}{12,434.3} \approx 0.138\). This is a weak positive correlation, but the initial handwritten equation suggests a strong correlation (m=4), so maybe there's a mistake in the data interpretation.

Part C: Solve for \(f(250)\) using the handwritten equation \(y = 4.00x - 222.00\) (assuming x is calories, y is sodium)

Substitute \(x = 250\) into the equation:
\(y = 4.00*250 - 222.00 = 1000 - 222 = 778\)

Problem 2

Part A: Write