Download Normal Distribution

WARM-UP
Determine whether the following are linear
transformations, combinations or both. Also find
the new mean and standard deviation for the
following.
1.
2.
3.
4.
5.
A = 2.5x
B=X+Y
C = X – 2y
D = -.5y
E = xy
Mean Standard
Deviation
X 16.5
2.5
Y 20
4.5
COUNTING…

Find the number of items in the sample space of
license plates containing 3 letters and 3 numbers
that can be repeated.
 What
if they can’t be repeated?
PERMUTATIONS

An arrangement of objects in a specific order
 Order Matters and No Repetitions
 EX:
How many ways can you arrange 3 people in
a picture?
EXAMPLE 2
 Suppose
a business owner has a choice of 5 locations.
She decides the rank them from best to worst
according to certain criteria. How many different
ways can she rank them?
 What
if she only wanted to rank the top 3?
PERMUTATION RULE!
n!
n Pr 
(n  r )!
 Where
need.
n = total # of objects and r = how many you
EXAMPLE 3
 A TV
news director wishes to use 3 news stories on
the evening news. She wants the top 3 out of 8
possible. How many ways can the program be set up?
COMBINATIONS
 A selection
of “n” objects without regard to order.
 When
different orderings of the same items are not
counted separately we have a combination problem.
 EX: AB is the same as BA
 When
different ordering of the same items are
counted separately, we have a permutation.
 EX: AB is different than BA
COMBINATION RULE
n!
n Cr 
(n  r )! r!
 Example1
: To survey opinions of customers at local
malls, a researcher decides to select 5 from 12. How
many ways can this be done?
EXAMPLE 2
 In
a club, there are 7 women and 5 men. A committee of 3
women and 2 men is to be chosen. How many different
possibilities are there?
 What
 At
about a committee of 5 with at least 3 women?
most 2 women?
BINOMIAL DISTRIBUTIONS
 Each

trial has only 2 possible outcomes
“success” or “failure”
 There
is a fixed # of trials (n)
 Trials are independent of each other
 The probability of a success (p) is constant
X ~ B(n, p)
– numerical probability of failure (1 – p)
 r – number of “successes” in n trials
q
BINOMIAL DISTRIBUTION FORMULA
 For


X ~ B(n, p) then
 n  r nr
P ( X  r )    p q
r
n
n!
   n Cr 
r!(n  r )!
r
n!
x
n x
 p (1  p )
x!(n  x)!
EXAMPLE 1
 A coin
is tossed 3 times. Find the probability of getting
exactly 2 heads.
EXAMPLE 2
 Public
Opinion reported that 5% of Americans are afraid
of being alone in the house at night. If a random sample
of 20 Americans is selected, find the probability that there
are exactly 5 people who are afraid of being along in the
house at night.
YOU TRY!
 A student
takes a random guess at 5 multiple choice
questions. Find the probability that the student gets
exactly 3 correct. Each question has 4 possible choices.
EXAMPLE 3
X
is binomially distributed with 6 trials and a probability
of success equal to 1/5 at each.
 What is the probability of at least one success?

Three or fewer successes?
EXAMPLE 2 REVISITED
 Public
Opinion reported that 5% of Americans are afraid
of being alone in the house at night. If a random sample
of 20 Americans is selected.
 Find the probability that at most 3 are afraid.

Find the probability that at least 3 are afraid.
YOU TRY AGAIN!
 A student
takes a random guess at 5 multiple choice
questions. Each question has 4 possible choices.
 Find the probability that the student gets at most 2
correct.

Find the probability that the student gets at least 2
correct.
MEAN & STANDARD DEVIATION
 For
a binomial distribution:
 p = probability of success and q = probability of failure
μ = p and σ = √(pq)  for 1 trial
μ = 2p and σ = √(2pq)  for 2 trials
μ = 3p and σ = √(3pq)  for 3 trials
In general… μ = np and σ = √(npq)  for n trials
EXAMPLE 1
 5%
of a batch of batteries are defective. A random sample
of 80 batteries is taken with replacement. Find the mean
and standard deviation of the number of defective
batteries in the sample.
2.
© 2011 Pearson Education, Inc
© 2011 Pearson Education, Inc
WARM UP
 A biased
coin is tossed 6 times. The probability of
heads on any toss is 0.3. Let X denote the number of
heads that come up.
 Calculate:
 P(X = 2)
 P(X < 3)
 P(1 < X < 5).
NORMAL DISTRIBUTION
A
normal distribution curve is symmetrical,
bell-shaped curve defined by the mean and
standard deviation of a data set.
 The
normal curve is a probability distribution
with a total area under the curve of 1.
CHARACTERISTICS OF A NORMAL DISTRIBUTION
 What
do the 3 curves have in common?
CHARACTERISTICS OF A NORMAL DISTRIBUTION
 The
curves may have different mean and/or standard
deviations but they all have the same characteristics
 Bell-shaped curve
 Symmetrical about the mean
 Mean, median and mode are the same


(not skew!)
Area under the curve is always 1 (100%)
STANDARD NORMAL DISTRIBUTION
 Written
as
Z ~ N(0, 1)
 Mean = 0 & Standard Deviation = 1
STANDARD NORMAL DISTRIBUTION
 Since
the total area under the curve is 1, we can
consider partial areas to represent probabilities.
Z-SCORES
A
standard normal distribution is the set of
all z-scores.
 All values can be transformed
from a normal distribution to
a standard normal by using
the z-score.
 It represents how many standard
deviations “x” is always from the mean.
 The z-score is positive if the data value
lies above the mean and negative if the
data value lies below the mean.
Z-SCORE EXAMPLES
 Suppose
SAT scores among college students are
normally distributed with a mean of 500 and a
standard deviation of 100. If a student scores a 700,
what would be their z-score?
MORE Z-SCORE EXAMPLES
 For
which test would a score of 78 have a higher
standing?
 A set of English test scores has a mean of 74 and a
standard deviation of 16.
 A set of math test scores has a mean of 70 and a
standard deviation of 8.
EVEN MORE Z-SCORE EXAMPLES
 What
will be the miles per gallon for a Toyota Camry
when the average mpg is 23, it has a z-value of 1.5
and a standard deviation of 5?
AREA WITH A TABLE
 Draw
the distribution curve
 Shade the area in which you are interested
 Use the table to find the areas

Might have to add or subtract to get what you want.
EXAMPLES FOR AREA
 Find
the area/probability of the following:
 Left of z = 1.99
P(z < 1.99)
 Left of z = 2.55
P(z < 2.55)
 Right of z = 1.11
P(z > 1.11)
MORE EXAMPLES FOR AREA
 Find
the area/probability of the following:
 Left of z = -2.50
P(z < -2.5)

Right of z = - 1.20
P(z > -1.2)
EVEN MORE EXAMPLES FOR AREA
 Find
the area/probability of the following:

P(0 < z < 2.32)

P(-1.2 < z < 2.3)
AND ONE MORE EXAMPLE FOR AREA
 Find

the area/probability of the following:
P(z < -3.01 and z > 2.43)
APPLICATION 1
A
Calculus exam is given to 500 students. The
scores have a normal distribution with a mean
of 78 and a standard deviation of 5. What
percent of the students have scores between
82 and 90? How many students have scores
between 82 and 90?
APPLICATION 2
A
Calculus exam is given to 500 students. The
scores have a normal distribution with a mean
of 78 and a standard deviation of 5. What
percent of the students have scores above
70? How many students scored above a 70?
APPLICATION 3
 Find
the probability of scoring below a 1400
on the SAT if the scores are normal
distributed with a mean of 1500 and a
standard deviation of 200.
FINDING Z-SCORES FROM AREA
 Find
the z-score above the mean with an area
to the left of z equal to 0.9325
 Find
the z-score below the mean with an area
to the left of z equal to 13.87%
MORE FINDING Z-SCORES FROM AREA
 Find
the z-score below the mean with an area
between 0 and z equal to 0.4066
EVEN MORE FINDING Z-SCORES FROM AREA
 Find
the z-score above the mean with an area
between 0 and z equal to 0.2123
 Find
the z to the right of the mean with an
area to the right of z equal to 0.0239
INVERSE NORMAL DISTRIBUTIONS

Find k for which P(x < k) = 0.95 given that x is normally
distributed with a mean of 70 and a standard deviation of
10.
APPLICATIONS

A professor determines that 80% of this year’s History
candidates should pass the final exam. The results are
expected to be normally distributed with a mean of 62 and
standard deviation of 13. Find the lowest score necessary to
pass the exam.
MORE APPLICATIONS

Researchers want to select people in the middle 60% of the
population based on their blood pressure. If the mean is 120
and the S.D. is 8. Find the upper and lower reading that
would qualify.
FINDING STATS BASED ON PROBABILITY

Sacks of potatoes with a mean weight of 5 kg are packed by
an automatic loader. In a test, it was found that 10% of bags
were over 5.2 kg. Use this information to find the standard
deviation of the process
MORE FINDING STATS BASED ON
PROBABILITY

Find the mean and the standard deviation of a normally
distributed random variables X, if P(x > 50) = 0.2 and P(x <
20) = 0.3