Stat 301 – HW 6 answers
1. chip failure time
a. 1 pt. -0.081 (rounded from -0.08066)
b. 2 pt. increasing the temperature by 10 degrees reduces time to failure by 55%
(or multiplies failure time by 0.446).
Note: calculated as change in log failtime = 10*slope = -0.806,
so multiplicative effect on failtime = exp(-0.806) = 0.446
c. 1 pt. -11.6 (rounded from -11.605)
d. 2 pt. increasing temperature by 10% reduces time to failure by 67%
(or multiplies failure time by 0.331).
Note: calculated as change in log failtime = log(1.1) * (-11.6) = 0.0953 * (-11.6) = -1.105,
so multiplicative effect on failtime is exp(-1.105) = 0.331
2. Voltage to separate oil from water
a. 2 pt. Variables with p-values less than 0.05 are Volume (p=0.0042), Salinity (p=0.0472) , and
Surfactant (p=0.0020)
b. 2 pt. F = 5.5, p = 0.0064. Very strong evidence that at least one coefficient is not zero.
(or reject the null hypothesis that all coefficients are zero).
c. 1 pt. All correlations between pairs of X variables are 0.
Note: There are non-zero correlations between Y and each X variable, but the question didn’t ask
about those.
d. 2 pt. When the correlations between X variables are 0, the simple regression slope is the same as
the multiple regression slope.
Note: In fact, the simple regression slope will be the same as the multiple regression slope for any
collection of X variables (e.g. a 3 variable or 4 variable model).
3.0
3. Voltage to separate oil from water, part 2.
a. 2 pt. The slope for Volume depends on the values of both salinity and surfactant. Many sketches
are possible. Here’s one, where each line is labelled as salinity, surfactant (e.g. 1,2 means the line
for salinity = 1, surfactant = 4). (Your plot should show at least two lines, with different slopes)
Voltage
1.0
1.5
2.0
2.5
1,2
4,4
0.0
0.5
1,4
40
50
60
70
Volume
80
90
b. 1 pt. The interaction model fits better: it has a smaller rMSE than does the first order model (the
one fit in problem 2).
Note: (optional because the problem doesn’t tell you to report the fitted model). That is:
E Voltage = -2.9 + 0.035 Volume + 0.64 Salinity + 1.18 Surfactant – 0.0078 Volume*Salinity – 0.012
Volume*Surfactant
c. 2 pt. Various answers acceptable. The minimum would be something like:
increasing Volume, Salinity or Surfactant increases the average Voltage. Increasing Salinity or
Surfactant decreases the slope for volume.
The best would be something like:
Each one unit increase in salinity increases voltage by 0.64 units and each one unit increase in
surfactant increases voltage by 1.18 units. The effect of a one unit increase in volume depends on
the salinity and surfactant concentration. At 0 salinity and 0 surfactant concentration, each one
unit increase in volume increases voltage by 0.035 units. Each one unit increase in salinity reduces
the volume slope by 0.0078 while each one unit increase in surfactant reduces the volume slope by
0.012.
d. 2 pt.
Salinity
0
3
3
Surfactant
0
0
3
Slope for Volume
0.035
0.012
-0.024
Calculated as:
0.035 – 0.0078*0 – 0.012*0
0.035 – 0.0078*3 – 0.012*0
0.035 – 0.0078*3 – 0.012*3