Help me please
The table shows the distance and costs of flights for a given airline with the same starting destination and different ending destinations.
Distance (miles)
1281 2796 790 150 3211 7810 812 210 7742 3578 1346 3214
Cost (dollars)
473 683 508 301 1115 3619 587 540 3013 1647 882 817
a. Find the equation of the median-median line for the data.
b. Predict the cost of a flight that travels a distance of 1950 miles.
To find the equation of the median-median line for the data, follow these steps:
Step 1: Sort the data pairs (distance, cost) in increasing order of distance.
Distance (miles) Cost (dollars)
150 301
210 540
790 508
812 587
1281 473
1346 882
2796 683
3211 1115
3214 817
3578 1647
7810 3619
7742 3013
Step 2: Calculate the median distance (Dm) and median cost (Cm).
To find the median, we need to sort the distances in ascending order:
150, 210, 790, 812, 1281, 1346, 2796, 3211, 3214, 3578, 7810, 7742
The median is the middle value, which in this case is the average of the two middle values: (1281 + 1346) / 2 = 1313.5.
So, Dm = 1313.5
To find the median cost, we need to sort the costs in ascending order:
301, 473, 508, 540, 587, 683, 817, 882, 1115, 1647, 3013, 3619
The median is the middle value, which is 817.
So, Cm = 817
Step 3: Calculate the slopes of the upper and lower lines.
The upper line passes through the points (Dm, Cm) and (maximum distance, maximum cost).
(Dm, Cm) = (1313.5, 817)
The maximum distance is 7742, and the corresponding cost is 3013.
So, the upper line passes through the points (1313.5, 817) and (7742, 3013).
The slope of the upper line (mU) is given by:
mU = (Cmax - Cm) / (Dmax - Dm)
= (3013 - 817) / (7742 - 1313.5)
= 2196 / 6428.5
= 0.3411 (approximately)
Similarly, the lower line passes through the points (minimum distance, minimum cost) and (Dm, Cm).
The minimum distance is 150, and the corresponding cost is 301.
So, the lower line passes through the points (150, 301) and (1313.5, 817).
The slope of the lower line (mL) is given by:
mL = (Cm - Cmin) / (Dm - Dmin)
= (817 - 301) / (1313.5 - 150)
= 516 / 1163.5
= 0.4444 (approximately)
Step 4: Find the y-intercept of the median-median line.
To find the y-intercept, substitute the slope and coordinates of the median (Dm, Cm) into the slope-intercept form equation: y = mx + b.
Using the upper line:
Cm = mU * Dm + b
817 = 0.3411 * 1313.5 + b
817 = 448.099 + b
b = 368.901 (approximately)
Using the lower line:
Cm = mL * Dm + b
817 = 0.4444 * 1313.5 + b
817 = 583.9294 + b
b = 233.0706 (approximately)
Step 5: Write the equation of the median-median line.
Since the median-median line is an average of the upper and lower lines, we can find its equation by taking the average of the slopes and y-intercepts.
Slope (m) = (mU + mL) / 2
= (0.3411 + 0.4444) / 2
= 0.39275 (approximately)
Y-intercept (b) = (bU + bL) / 2
= (368.901 + 233.0706) / 2
= 300.9858 (approximately)
The equation of the median-median line is:
C = 0.39275D + 300.9858
Now, to predict the cost of a flight that travels a distance of 1950 miles, substitute the distance (D) into the equation of the median-median line:
C = 0.39275 * 1950 + 300.9858
C ≈ 1737.875 + 300.9858
C ≈ 2038.8618 (approximately)
Therefore, the predicted cost of a flight that travels a distance of 1950 miles is approximately $2038.86.