Stats Question help?

Question

The least squares regression line is calculaed to be

y = 1428 - 1.37x

It is determined that 86% if the variation in the wind speed of a cyclone can be accounted for by the atmospheric pressure.

a) provide an interpretation of the slope of the least squares regression line.

b) what is the value of the correlation between atmospheric pressure and wind speed?

c) for one cyclone in the sample, the atmospheric pressure was 1000 millibars and the wind speed was 43 mph. What is the value of the residual for this cyclone?

d) Given the strong correlation, can we conclude that lower atmospheric pressure causes the wind speed in cyclone to increase? why or why not?

Answer 1

When you are given "86% of the variation in the wind speed of a cyclone can be accounted for by the atmospheric pressure" you can learn several things.

1) The dependent variable (aka "y-value") is wind speed (of a cyclone).

2) The independent variable (aka "x-value") is atmospheric pressure.

3) The coefficient of determination or r-squared (r²) is 0.86

a) The slope of a linear equation is the change in y divided by the change in x. In this case, that is the change in wind speed per change in atmospheric pressure. In other words, in this case, as atmospheric pressure increases by a millibar, wind speed decreases by 1.37 mph (I got the units from question c)

b) If r² = 0.86 then r = ±√0.86 ≈ ± 0.927 and, in this case, because the slope of the regression line is negative, the r-value or correlation coefficient is approximately -0.927.

c) Residual = Actual Value - Predicted Value

Using the regression line y = 1428 - 1.37x and plugging in x = 1000, we get y = 58. So the prediction for a pressure of 1000 millibars is a wind speed of 58 mph. The actual wind speed is 43. So the residual is 43 - 58 = -15

d) From the regression line, we see that the lower the pressure, the higher the wind speed so "yes". The reason is the negative correlation.