Introduction to Probability and Statistics for Engineers and Scientists

Problems 413

that each animal has a random threshold and will become ill if the dosage level exceeds
that threshold. The exponential distribution has sometimes been used as the threshold
distribution. For instance, a model considered in Freedman and Zeisel (“From Mouse to
Man: The Quantitative Assessment of Cancer Risks,”Statistical Science, 1988, 3 , 1, 3–56)
supposes that a mouse exposed toxunits of DDT (measured in ppm) will contract cancer
of the liver with probability

p(x)= 1 −e−ax, x> 0

Becauseofthelackofmemoryoftheexponentialdistribution, thisisequivalenttoassuming
that if the mouse who is still healthy after receiving a (partial) dosage of levelxis as good
as it was before receiving any dosage.
It was reported in Freedman and Zeisel that 84 of 111 mice exposed to DDT at a level
of 250 ppm developed cancer. Therefore,αcan be estimated from

1 −e−^250 αˆ=

84

111

or

αˆ=−

log(27/111)

250

=.005655 ■

Problems.........................................................

1. The following data relatex, the moisture of a wet mix of a certain product, toY,
   the density of the finished product.

xi Yi

5 7.4

6 9.3

7 10.6

10 15.4

12 18.1

15 22.2

18 24.1

20 24.8

(a) Draw a scatter diagram.

(b) Fit a linear curve to the data.