Written homework problem 11
Assigned 10/2 and due 10/7
(This problem was from the second take-home exam when I taught Math 113 in Fall 2014.)
A statistician is trying to determine precisely the optimal amount of ice cream to eat. They
have gathered data from a number of people about how much ice cream they ate and how they
felt afterwards. The data is given below.
i Ice cream eaten (in fluid ounces) xi Goodness of feeling yi
1
16
18
2
5
8
3
8
20
4
10
17
5
25
13
6
40
-42
7
30
5
8
17
23
9
2
10
12
18
10
The statistician believes that this data can be accurately modeled by a parabola y = β0 + β1 x +
β2 x2 , where x is the quantity of ice cream eaten, and y is the goodness of feeling.
Find the choice of β0 , β1 and β2 that best fits the data given, that is, the one that minimizes
the function
10
X
2
SSE(β0 , β1 , β2 ) =
yi − (β0 + β1 xi + β2 x2i ) .
i=1
You may assume that a minimum exists. Based on the values of β0 , β1 and β2 obtained, what
is the optimal quantity of ice cream to eat (on one occasion)? [In order to do this, you will
need to solve a system of three linear equations in three unknowns. Also, the numbers on this
problem don’t work out particularly nicely.]
1