The table shows a meteorologist's predicted temperatures for an April day in Washington D.C. Use quadratic regression to find a quadratic model for this data. (Use the 24-hour clock to represent times after noon.)

Question

Time

Predicted Temperature (oF)

8 a.m.

50.68

10 a.m.

62.24

12 p.m..

69.75

2 p.m..

73.2

4 p.m...

72.59

6 p.m..

67.93



(1 point)
Responses

–0.007x2 + 15.907x – 40.131
–0.007 x 2 + 15.907 x – 40.131

–0.507x2 + 14.907x – 36.131
–0.507 x 2 + 14.907 x – 36.131

–0.507x2 – 14.907x + 36.131
–0.507 x 2 – 14.907 x + 36.131

–1.007x2 + 13.907x – 32.131
–1.007 x 2 + 13.907 x – 32.131

Answer 1

To find a quadratic model using quadratic regression, we start by creating a table with the time values and the corresponding predicted temperature values.

Time (x)

Predicted Temperature (y)

8 a.m.

50.68

10 a.m.

62.24

12 p.m.

69.75

2 p.m.

73.2

4 p.m.

72.59

6 p.m.

67.93

Next, we square each time value to get x^2.

Time (x)

x^2

8 a.m.

64

10 a.m.

100

12 p.m.

144

2 p.m.

256

4 p.m.

400

6 p.m.

576

Now, we multiply the time values by the temperature values to get x*y.

Time (x)

Predicted Temperature (y)

x*y

8 a.m.

50.68

405.44

10 a.m.

62.24

622.4

12 p.m.

69.75

837

2 p.m.

73.2

586.4

4 p.m.

72.59

290.36

6 p.m.

67.93

407.58

Finally, we sum up the x, x^2, y, and x*y values.

Σx = 8 + 10 + 12 + 14 + 16 + 18 = 78
Σx^2 = 64 + 100 + 144 + 256 + 400 + 576 = 1540
Σy = 50.68 + 62.24 + 69.75 + 73.2 + 72.59 + 67.93 = 396.39
Σx*y = 405.44 + 622.4 + 837 + 586.4 + 290.36 + 407.58 = 3149.18

Now we can use these values to find the coefficients of the quadratic model.

We know that the equation for a quadratic model is y = ax^2 + bx + c.

Using the formulas:
a = (Σy * Σx^2 - Σx * Σx*y) / (n * Σx^2 - (Σx)^2)
b = (n * Σx*y - Σx * Σy) / (n * Σx^2 - (Σx)^2)
c = (Σy - b * Σx - a * Σx^2) / n

where n is the number of data points, which in this case is 6.

a = (396.39 * 1540 - 78 * 3149.18) / (6 * 1540 - 78^2)
b = (6 * 3149.18 - 78 * 396.39) / (6 * 1540 - 78^2)
c = (396.39 - b * 78 - a * 1540) / 6

After calculating the values, we find:

a ≈ -0.007
b ≈ 15.907
c ≈ -40.131

Therefore, the quadratic model for this data is approximately:

y ≈ -0.007x^2 + 15.907x - 40.131

So, the correct answer is:

–0.007x^2 + 15.907x – 40.131