Standardising Normal Distributions

1
Standardising Normal Distributions
We are interested in worms so we look on a
website and it tells us that the lengths of worms
are normally distributed with a mean of 21cm and
a standard deviation of 1.5cm. Our natural
curiosity leads us to ask.. I wonder what the
probability is of randomly finding a worm that is
longer than 23cm?
We condense all this gibberish to
Worms
Standard deviation squared (variance)
Follow
Normal distribution
Mean
We want to find
We dont have a table with results for this
distribution, but we do have a table of results
for
We shift our distribution along and stretch or
shrink it to make it the same as the standard
normal distribution
2
0
21
Standard Normal Curve
Length of worms
The length of worms has a greater standard
deviation than the standard normal curve so it is
fatter
To change the X distribution into the Z
distribution we must go backwards 21 and then
divide by the standard deviation
The magic formula
3
Back to the question
DIAGRAM!
4
Another example We are still looking at our
worms
What is the probability that a worm found at
random is less than 21.5cm long?
5
A) P(Xlt164.08) B) P(Xgt157.04) C) P(Xgt140.1) D)
P(Xlt148.24) E) P(146.7ltXlt157.7) F)
P(141.2ltXlt196.2)
0.7389 0.3745 0.6736 0.4681 0.1964 0.6554
6
Contextual Questions
The length of time patients have to wait in Dr
Booths waiting room is known to be normally
distributed with mean 14 minutes, and standard
deviation 4 minutes. A) Find the probability
that I will have to wait more than 20 minutes to
see the doctor B) What proportion of patients
wait less than 10 minutes?
7
Packets of breakfast cereal are said to contain
550g. The manufacturer knows that the weights
are normally distributed with mean 551.2g, and
standard deviation 15g. What proportion of
packets will contain more than the stated weight?
A biologist has studied a particular tropical
insect and she has discovered that its lifespan
is normally distributed. The mean lifespan of
the insect is 72 days and the standard deviation
of its lifespan is eight days. Find the
probability that the next insect studied
lives A) Fewer than 70 days B) more than 76
days C) between 68 and 78 days