Transcript Characteristics of a Normal Probability Distribution

STATISTICS
“The Normal Probability Distribution”
11.0 The Normal Probability Distribution
11.0 The Normal Probability Distribution
Characteristics of a Normal
Probability Distribution
– A continuous random variables and a continuous
probability distribution is known as the normal distribution
– A Normal Process has the following characteristics:
1) The mean, median and mode are the same value
2) The distribution is bell shaped and symmetrical around the mean.
3) The total area under the curve is equal to 1
4) The left and right tails of the normal probability distribution extend
indefinitely, never quite touching the horizontal axis
11.0 The Normal Probability Distribution
Characteristics of a Normal
Probability Distribution
– A smaller standard deviation results in a “skinner” curve
that’s tighter and taller around the mean.
– A larger standard deviation makes for a “fatter” curve
that’s more spread out and not as tall.
– The value of mean and standard deviation, completely
describe the shape of the distribution.
11.0 The Normal Probability Distribution
The Normal Probability Distribution
– To make Normal Probability Distribution, we need
to define the standard normal distribution, which
is a normal distribution with a μ=0 and σ=1.
– This standard normal distribution is the basis for all
normal probability calculation:
z=x-μ
z = the number of difference between x and μ, known as
σ
the standard z-score
11.0 The Normal Probability Distribution
The Normal Probability Distribution
– Once obtain the z-score, use the Standard
Normal Table to determine the probability.
– In general, you can use the following two
relationships for any value a when dealing with
negative z-score:
P[z >-a] = P[z ≤+a]
P[z≤-a] = 1 – P[z ≤+a]
11.0 The Normal Probability Distribution
• TRY THIS!!!
1) The amount of toxic spray use to kill Japanese beetle used
each year follows a normal distribution with a mean of 60
liter and a standard deviation of 5 liter. What is the
probability:
a) Less than 64.3 liter (Answer: 0.8051)
b) More than 62.5 liter (Answer: 0.3085)
c) More than 54 liter (Answer: 0.8849)
2) The speed of cars passing through a checkpoint follows a
normal distribution with μ = 62.6 m/h and σ = 3.7 m/h What
is the probability of the next car passing will:
a) Be exceeding 65.5 m/h (Answer: 0.2177)
b) Be exceeding 58.1 m/h (Answer: 0.888)
c) Be between 61 and 70 m/h (Answer: 0.6436)
Exercises
The lengths of steel beams made by a particular steel mill is normally.
Distributed with a mean of 10.35 metres and a standard deviation of 2.25
metres.
a) Find the probability that the length of a steel beam will be over 10.86 metres.
b) Find the probability that the length of a steel beam will be over 11.40 metres.
c)
Find the probability that the length of a steel beam will be over 9.41 metres.
d) For a particular application, any beam less than 9.05 metres must be scrapped. What
percentage of beams would expect to be scrapped?
e) Find the probability that the length of a steel beam will less than 10.96 metres.
f)
Find the probability that the length of a steel beam will be between 10.1 and 11.10
metres.
g) Find the probability that the length of a steel beam will be between 10.86 and 11.05
metres.
h) Find the probability that the length of a steel beam will be between 9.01 and 10.08
metres.