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Appendix 4.A.1 Minitab Applications
279
Areas under normal curves between points a and b are calculated by subtracting FY (a) from FY (b), just as we did in Section 4.3 (recall the Comment
after Definition 4.3.1). There is no need, however, to reexpress the probability
as an area under the standard normal curve. Figure 4.A.1.2 shows the Minitab
calculation for the probability that the random variable Y lies between 48 and
51, where Y is normally distributed with μ = 50 and σ = 4. According to the
computer,
P(48 < Y < 51) = FY (51) − FY (48)
= 0.598706 − 0.308538
= 0.290168
Figure 4.A.1.2
MTB > cdf 51;
SUBC> normal 50 4.
Cumulative Distribution Function
Normal with mean = 50 and standard deviation = 4
x
P( X <= x)
51
0.598706
MTB > cdf 48;
SUBC> normal 50 4.
Cumulative Distribution Function
Normal with mean = 50.0000 and standard deviation = 4.00000
x
P( X <= x)
48
0.308538
MTB > let k1 = 0.598706 − 0.308538
MTB > print k1
Data Display
k1
0.290168
On several occasions in Chapter 4 we made use of Minitab’s RANDOM command, a subroutine that generates samples from a specific pdf. Simulations of that
sort can be very helpful in illustrating a variety of statistical concepts. Shown in
Figure 4.A.1.3, for example, is the syntax for generating a random sample of size 50
from a binomial pdf having n = 60 and p = 0.40. And calculated for each of those
fifty observations is its Z ratio, given by
X − 60(0.40)
X − 24
X − E(X )
=√
= √
Z -ratio = √
Var(X )
60(0.40)(0.60)
14.4
[By the DeMoivre-Laplace theorem, of course, the distribution of those ratios
should have a shape much like the standard normal pdf, f Z (z).]
Figure 4.A.1.3
MTB > random 50 c1;
SUBC> binomial 60 0.40.
MRB > print c1
Data Display
C1
27 29 23 22 21 21 22 26 26 20 26 25 27
32 22 27 22 20 19 19 21 23 28 23 27 29
13 24 22 25 25 20 25 26 15 24 17 28 21
16 24 22 25 25 21 23 23 20 25 30
MTB > let c2 = (c1 - 24)/sqrt(14.4)
MTB > name c2 ’Z-ratio’
MTB > print c2
Data Display
Z-ratio
0.79057 1.31762 −0.26352 −0.52705 −0.79057 −0.79057
0.52705 0.52705 −1.05409 0.52705 0.26352 0.79057
−0.52705 0.79057 −0.52705 −1.05409 −1.31762 −1.31762
−0.26352 1.05409 −0.26352 0.79057 1.31762 −2.89875
−0.52705 0.26352 0.26352 −1.05409 0.26352 0.52705
0.00000 −1.84466 1.05409 −0.79057 −2.10819 0.00000
0.26352 0.26352 −0.79057 −0.26352 −0.26352 −1.05409
1.58114
−0.52705
2.10819
−0.79057
0.00000
−2.37171
−0.52705
0.26352