Page 241 - Matrix Analysis & Applied Linear Algebra

236 Chapter 4 Vector Spaces

4.6.6. After studying a certain type of cancer, a researcher hypothesizes that
in the short run the number (y) of malignant cells in a particular tissue
grows exponentially with time (t). That is, y = α 0 e α 1 t . Determine least
squares estimates for the parameters α 0 and α 1 from the researcher’s
observed data given below.
t (days) 1 2 3 4 5

y (cells) 16 27 45 74 122
Hint: What common transformation converts an exponential function
into a linear function?

4.6.7. Using least squares techniques, fit the following data

x −5 −4 −3 −2 −1 0 1 2 3 4 5

y 2 7 9 12 13 14 14 13 108 4
with a line y = α 0 + α 1 x and then fit the data with a quadratic y =
2
α 0 +α 1 x+α 2 x . Determine which of these two curves best fits the data
by computing the sum of the squares of the errors in each case.

4.6.8. Consider the time (T) it takes for a runner to complete a marathon (26
miles and 385 yards). Many factors such as height, weight, age, previous
training, etc. can influence an athlete’s performance, but experience has
shown that the following three factors are particularly important:
height (in.)
x 1 = Ponderal index = ,
1
[weight (lbs.)] 3
x 2 = Miles run the previous 8 weeks,
x 3 = Age (years).
A linear model hypothesizes that the time T (in minutes) is given by
T = α 0 + α 1 x 1 + α 2 x 2 + α 3 x 3 + ε, where ε is a random function
accounting for all other factors and whose mean value is assumed to
be zero. On the basis of the five observations given below, estimate the
expected marathon time for a 43-year-old runner of height 74 in., weight
180 lbs., who has run 450 miles during the previous eight weeks.
T x 1 x 2 x 3
181 13.1 619 23
193 13.5 803 42
212 13.8 207 31
221 13.1 409 38
248 12.5 482 45
What is your personal predicted mean marathon time?