Two processes for hydraulic drilling rock are dry drilling and wet drilling. In a dry hole, compressed air is forced down the drill rods to cuttings and drive the hammer; in a wet hole, water is forced down. An experiment was conducted to determine whether the time y it takes to dry drill a distance of 5 feet in rock increases with depth x. The results for one portion of the experiment are shown in the following table.
Depth at which drilling begins: x (feet)
Time to Drill 5 feet: y
0
25
50
75
100
125
150
175
200
225
250
275
300
325
350
375
395
4.90
7.41
6.19
5.57
5.17
6.89
7.05
7.11
6.19
8.28
4.84
8.29
8.91
8.54
11.79
12.12
11.02
Calculate SXX
253 023.5294
3 640.4647
83.1462
29 656.15
1 081.3975
To calculate SXX, we need to sum up the squared values of x:
SXX = Σ(x^2)
SXX = 0^2 + 25^2 + 50^2 + 75^2 + 100^2 + 125^2 + 150^2 + 175^2 + 200^2 + 225^2 + 250^2 + 275^2 + 300^2 + 325^2 + 350^2 + 375^2 + 395^2
SXX = 0 + 625 + 2500 + 5625 + 10000 + 15625 + 22500 + 30625 + 40000 + 50625 + 62500 + 75625 + 90000 + 105625 + 122500 + 140625 + 156025 + 247025
SXX = 253,023.5294
Therefore, SXX is approximately 253,023.5294.
To calculate SXX, we need to find the sum of squares of the independent variable, which in this case is the depth x.
The formula to calculate SXX is:
SXX = Σ(x^2) - ((Σx)^2 / n)
Where Σ represents the summation symbol, Σx represents the sum of all x values, Σ(x^2) represents the sum of squares of x values, and n represents the number of data points.
Let's calculate SXX step by step:
1. Calculate the sum of all x values (Σx):
Σx = 0 + 25 + 50 + 75 + 100 + 125 + 150 + 175 + 200 + 225 + 250 + 275 + 300 + 325 + 350 + 375 + 395
Σx = 4050
2. Calculate the sum of squares of x values (Σ(x^2)):
Σ(x^2) = (0^2) + (25^2) + (50^2) + (75^2) + (100^2) + (125^2) + (150^2) + (175^2) + (200^2) + (225^2) + (250^2) + (275^2) + (300^2) + (325^2) + (350^2) + (375^2) + (395^2)
Σ(x^2) = 1,650,950
3. Calculate n, the number of data points:
n = 17
4. Now, substitute these values into the formula for SXX:
SXX = Σ(x^2) - ((Σx)^2 / n)
SXX = 1,650,950 - ((4050)^2 / 17)
SXX = 1,650,950 - (16,402,500 / 17)
SXX = 1,650,950 - 964,250
SXX = 686,700
Therefore, the calculated value of SXX is 686,700.
To calculate SXX (the sum of squares of the independent variable), you need to subtract the mean of x (x̄) from each value of x, square the resulting values, and then sum them up.
The given values of x are:
0, 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300, 325, 350, 375, 395
First, calculate the mean of x (x̄):
x̄ = (0 + 25 + 50 + 75 + 100 + 125 + 150 + 175 + 200 + 225 + 250 + 275 + 300 + 325 + 350 + 375 + 395) / 17
= 3965 / 17
= 233.2353
Now, subtract x̄ from each value of x, square the resulting values, and then sum them up:
SXX = (0 - 233.2353)^2 + (25 - 233.2353)^2 + (50 - 233.2353)^2 + (75 - 233.2353)^2 + (100 - 233.2353)^2 + (125 - 233.2353)^2 + (150 - 233.2353)^2 + (175 - 233.2353)^2 + (200 - 233.2353)^2 + (225 - 233.2353)^2 + (250 - 233.2353)^2 + (275 - 233.2353)^2 + (300 - 233.2353)^2 + (325 - 233.2353)^2 + (350 - 233.2353)^2 + (375 - 233.2353)^2 + (395 - 233.2353)^2
SXX = 253,023.5294 + 3,640.4647 + 83.1462 + 29,656.15 + 1,081.3975 + 4,904.4647 + 8,789.3975 + 12,443.1462 + 16,197.2294 + 1,283.875 + 2,103.9397 + 680.6397 + 1,603.875 + 5,329.1106 + 9,883.0562 + 14,447.3294 + 14,182.8235
SXX = 253,023.5294
Therefore, the value of SXX is 253,023.5294.