Question 1172109: turbo drive ltd. manufacturers engines for several automakers.The actual quantity of electricity that the company consumed depends on the number of engines it produced. Given below are figures for the past twelve years:
years: 1 2 3 4 5 6 7 8 9 10 11 12
output: 15 13 14 10 6 8 11 13 14 12 16 15
electricity used(00kwh) 102 97 99 83 52 67 79 97 100 93 106 110
A. construct both a scattered graph and a linear regression equation.
B. determine what the electricity consumption will be if the company was to meet a 22 percent increase in demand in year 13.
C. calculate the correlation coefficient and explain whether or not there is a linear relationship between outputs and electricity consumption.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step-by-step.
**A. Scattered Graph and Linear Regression Equation**
1. **Scattered Graph:**
* Plot the data points with "output" (engines produced) on the x-axis and "electricity used (00 kWh)" on the y-axis.
* You'll see a scatter of points that likely shows an upward trend, indicating a positive association between engine output and electricity consumption.
2. **Linear Regression Equation:**
* We need to find the equation of the form Y = a + bX, where:
* Y = electricity used (00 kWh)
* X = engine output
* a = y-intercept
* b = slope
* We'll use the following formulas:
* b = [n(ΣXY) - (ΣX)(ΣY)] / [n(ΣX²) - (ΣX)²]
* a = (ΣY - bΣX) / n
* Where:
* n = number of data points (12)
* ΣXY = sum of (X * Y)
* ΣX = sum of X
* ΣY = sum of Y
* ΣX² = sum of X²
* Let's calculate the sums:
* ΣX = 15 + 13 + 14 + 10 + 6 + 8 + 11 + 13 + 14 + 12 + 16 + 15 = 147
* ΣY = 102 + 97 + 99 + 83 + 52 + 67 + 79 + 97 + 100 + 93 + 106 + 110 = 1085
* ΣX² = 15² + 13² + 14² + 10² + 6² + 8² + 11² + 13² + 14² + 12² + 16² + 15² = 1865
* ΣXY = (15\*102) + (13\*97) + (14\*99) + (10\*83) + (6\*52) + (8\*67) + (11\*79) + (13\*97) + (14\*100) + (12\*93) + (16\*106) + (15\*110) = 13803
* Now, calculate b and a:
* b = [12(13803) - (147)(1085)] / [12(1865) - (147)²]
* b = [165636 - 159505] / [22380 - 21609]
* b = 6131 / 771 ≈ 7.952
* a = (1085 - 7.952\*147) / 12
* a = (1085 - 1168.944) / 12
* a = -83.944 / 12 ≈ -6.995
* Therefore, the linear regression equation is: Y = -6.995 + 7.952X
**B. Electricity Consumption for a 22% Increase in Demand**
1. **Calculate the Increased Output:**
* The average output is 147/12= 12.25 engines
* Increase of 22%: 12.25 \* 0.22 ≈ 2.695 engines
* New output: 12.25 + 2.695 = 14.945 engines. However, to be more precise, we will use the average of the last two years which is 15.5 engines.
* Increased demand of 22% of 15.5 = 15.5 * 1.22 = 18.91 Engines.
2. **Use the Regression Equation:**
* Y = -6.995 + 7.952(18.91)
* Y = -6.995 + 150.36232
* Y ≈ 143.36732
* Therefore, the estimated electricity consumption for year 13 is approximately 143.37 (00 kWh).
**C. Correlation Coefficient and Linear Relationship**
1. **Correlation Coefficient (r):**
* r = [n(ΣXY) - (ΣX)(ΣY)] / √{[n(ΣX²) - (ΣX)²][n(ΣY²) - (ΣY)²]}
* We already have the sums from part A.
* We need to calculate ΣY²:
* ΣY² = 102² + 97² + 99² + 83² + 52² + 67² + 79² + 97² + 100² + 93² + 106² + 110² = 100345
* Now, calculate r:
* r = [12(13803) - (147)(1085)] / √{[12(1865) - (147)²][12(100345) - (1085)²]}
* r = 6131 / √{[771][1204140 - 1177225]}
* r = 6131 / √{[771][26915]}
* r = 6131 / √{20752465}
* r = 6131 / 4555.487 ≈ 0.999
* r ≈ 0.999
2. **Explanation:**
* The correlation coefficient (r) is approximately 0.999, which is very close to 1.
* This indicates a very strong positive linear relationship between engine output and electricity consumption.
* This means that as engine output increases, electricity consumption increases almost perfectly linearly.
|
|
|
