Question

TRUE or FALSE

(a) In a scatterplot, the response is shown on the horizontal axis with the explanatory variable
on the vertical axis.
(b) If all of the data lie along a single line with nonzero slope, then the r
2 of the regression is
1. (Assume the values of the explanatory variable are not identical).
(c) If the correlation between the explanatory variable and the response is zero, then the slope
will also be zero.
(d) The use of a linear equation to describe the assocation between price and sales implies
that we expect equal differences in sales when comparing periods with prices $10 and $11
and periods with prices $20 and $21.
(e) The linear equation (estimated from a sequence of daily observerations):
Estimated shipments = b0 + 0.9*(Orders Processed) implies that we expect twice as many
shipments when the number of orders processed doubles only if b0 = 0.
(f) The intercept estimates how much the response changes on average with changes in the
predictor.

Answer 1

I'll do parts (a) through (c) to get you started

(a) False. The explanatory variable is always along the x axis, and the response variable is along the y axis. The response variable reacts to whatever the input explanatory variable is, hence the name "response".

---------

(b) True. If all data lie along a single straight line with nonzero slope, then either r = -1 or r = 1. If we have all data points on a line with some negative slope, then r = -1. If the regression slope is positive, then r = 1. Either way, it will lead to r^2 = 1 because (1)^2 = 1 and (-1)^2 = 1. In other words, there are two solutions to r^2 = 1 and they are r = -1 or r = 1. In the case of r = -1, we say the data has perfect negative linear correlation. With r = 1, the data has perfect positive linear correlation.

---------

(c) True. If r > 0, then we have some level of positive correlation which means the regression slope is positive in some way. If r < 0, then the regression slope is negative in some way. That must mean r = 0 leads to the regression slope having a flat horizontal line. This could mean one of two things: i) The points are all on or near the same curve such as a parabola, ii) The points are completely randomly scattered about. If case (i) happens, then it's better to use nonlinear regression instead of linear regression.