Question

Windmills generate electricity by transferring energy from wind to a turbine

Answer 1

The complete question is reported in the attached picture.

1) The regression analysis was performed by the computer, in the output we have the coefficients we need: 0.137 and 0.240.
Therefore the equation of the least squares regression line will be:

`\hat{y} = 0.137 + 0.240x`

where:

`\hat{y}` = estimate of electricity production [A]
x = wind speed [mph]

2) Let's apply the previous formula to our cases:
x = 25 -->
`\hat{y}` = 0.137 + 0.240(25) = 6.137 A
x = 15 -->
`\hat{y}` = 0.137 + 0.240(15) = 3.737 A

The difference is 6.137 - 3.737 = 2.4 A

The windmill would be expected to produce 2.4 A more when the wind speeds at 25mph with respect to when the wind speeds at 15 mph.

3) The proportion of variation in electricity production explained by the linear relationship with the wind speed is given by the R-Sq value, which is 0.873.

Hence, the linear relationship between electricity production and wind speed explains the 87.3% of the variation in production.

4) The text says that the assumptions were checked. The computer performed all the calculations, finding as t-score for wind speed:
T = Coef / SE Coef = 0.240 / 0.019 = 12.63
which was converted to a p-value:
P = 0.000

Therefore, being p <0.05, the null hypothesis can be rejected and we can say that electricity production by the windmill is related to wind velocity.