During an experiment conducted on the effort E to lift a load W when using a simple the following values were recorded.
………………………………………………………………………………………………
|Effort E| 20 | 24.8|30 | 35 | 45 |
……………………………………………………………………………………………
|Load W |100 | 200 |300 | 400| 600|
|…………………………………………………………………………………………
If E and W are connected by the law of the form E= aw+b, determine.
(a).The value of a and b .
(b).the value of E when W = 280N.
To determine the values of 'a' and 'b' in the equation E = aw + b, we need to use the given data points. Since we have multiple data points, we can use the method of linear regression to find the best line that fits these data points.
Linear regression is a statistical analysis technique used to find the relationship between two variables. In this case, we want to find the relationship between the effort 'E' and the load 'W'.
Here are the steps to find the values of 'a' and 'b':
Step 1: Calculate the means of E and W.
- Calculate the mean (average) of the effort values:
mean_E = (20 + 24.8 + 30 + 35 + 45) / 5.
- Calculate the mean of the load values:
mean_W = (100 + 200 + 300 + 400 + 600) / 5.
Step 2: Calculate the deviations from the mean.
- For each data point, calculate the deviation of E and W from their respective means.
deviation_E = (E - mean_E), deviation_W = (W - mean_W).
Step 3: Calculate the sum of the products of deviations.
- Calculate the sum of the products of deviation_E and deviation_W for each data point.
sum_of_products = Σ(deviation_E * deviation_W).
Step 4: Calculate the sum of the squared deviations.
- Calculate the sum of the squared deviations of E for each data point.
sum_of_squared_deviations_E = Σ(deviation_E^2).
Step 5: Calculate the slope 'a' and the intercept 'b'.
- The slope 'a' can be calculated as:
a = sum_of_products / sum_of_squared_deviations_E.
- The intercept 'b' can be calculated as:
b = mean_W - (a * mean_E).
Step 6: Substitute the values of 'a' and 'b' into the equation.
- E = a * W + b.
Now let's calculate the values of 'a' and 'b' using the given data:
Mean of E = (20 + 24.8 + 30 + 35 + 45) / 5 = 30.16.
Mean of W = (100 + 200 + 300 + 400 + 600) / 5 = 320.
Deviation of E:
20 - 30.16 = -10.16,
24.8 - 30.16 = -5.36,
30 - 30.16 = -0.16,
35 - 30.16 = 4.84,
45 - 30.16 = 14.84.
Deviation of W:
100 - 320 = -220,
200 - 320 = -120,
300 - 320 = -20,
400 - 320 = 80,
600 - 320 = 280.
Sum of the products of deviations (-10.16 * -220) + (-5.36 * -120) + (-0.16 * -20) + (4.84 * 80) + (14.84 * 280) = 61796.76.
Sum of the squared deviations:
(-10.16)^2 + (-5.36)^2 + (-0.16)^2 + (4.84)^2 + (14.84)^2 = 761.2504.
Slope 'a':
a = 61796.76 / 761.2504 = 81.136.
Intercept 'b':
b = 320 - (81.136 * 30.16) = -102.07104.
Therefore, the values of 'a' and 'b' in the equation E = aw + b are:
a ≈ 81.136
b ≈ -102.07104
To find the value of E when W = 280N, we can substitute the value of W into the equation E = aw + b:
E = 81.136 * 280 - 102.07104
E ≈ 22715.69856
Therefore, when W = 280N, E ≈ 22715.69856.