Lecture 10.4 bt

Today’s Agenda
 Attendance / Announcements
 Sections 10.4

Population vs. Sample
 Sample Mean:
 Sample Standard Deviation:
 Population Mean:
 Population Standard Deviation:
x
s



Analyzing Real World Data
Below are the scores for the Anatomy and
Physiology Final Exam (30 students)
79 51 67 50 78 62
89 83 73 80 88 48
60 71 79 89 63 55
98 71 40 81 46 50
61 61 50 90 75 61

Continuous Probability Distributions
 Distributions for continuous random variables
 Usually the result of measurement:
 Height, time, distance,…
 Usually concerned with the percentage of population
(probability) within a certain range
 This is because a continuous random variable has an
infinite amount of values within any range, so we don’t
think in terms of probability for a specific value.

The Normal distribution
 Considered one of the most important distribution in all
of statistics.
 We’ve seen the idea of a “bell shaped and symmetric
curve.” This is the normal distribution……

Standardizing Normal Curve
 The standardized (or normalized) z-score
is basically “how many standard
deviations the value is from the mean”



x
z

The Normal Curve
The following are synonymous when
it comes to the normal curve:
• Find the area under the curve …
• Find the percentage of the population …
• Find the probability that …

Using a z-Table to find probabilities
Note: Our z-table only gives area to the left
(or probabilities less than z)

Find Probability
that z < 0.97
Z-scores: -2 -1 0 1 2 3-3
z = 0.97
Find area under
the curve to the
left of z = 0.97
)97.0( zP
Using a Z-Table to find probabilities

Find Probability
that z < 0.97
Since z > 0, use
positive side

Find Probability
that z < -2.91
Z-scores: -2 -1 0 1 2 3-3
-2.91
Find area under the
curve to the left of
z = -2.91

Find Probability
that z < -2.91
* Since z < 0, use
negative side

 Not all Z-Tables are alike!

 But we can still use our z-table to
find areas to the right (probability
greater than), as well as areas
between two values (probability
between two values).

Find Probability
that z > 0.75
Z-scores: -2 -1 0 1 2 3-3
0.75
Find area under the
curve to the right
of z = 0.75
)75.0( zP
Finding Area to the Right

Finding Area to the Right
 Two Methods
 Using the Complement
 Using Symmetry

Complement Method
Z-scores: -2 -1 0 1 2 3-3
0.75
Find area under the
curve to the right
of z = 0.75
Find Probability
that z > 0.75
)75.0( zP

Complement Method
- Use fact that
area under entire
curve is 1.
- And that we
can find area to
the left
Z-scores: -2 -1 0 1 2 3-3
0.75
1)75.0()75.0(  zPzP
Get
from
table Unknown

Complement Method
Z-scores: -2 -1 0 1 2 3-3
0.75
Find area under the
curve to the right
of z = 0.75
Find Probability
that z > 0.75
7734.0)75.0( zP

Find Probability
that z > 0.75
Z-scores: -2 -1 0 1 2 3-3
0.75
Find area under the
curve to the right
of z = 0.75
Complement Method
)75.0(1)75.0(  zPzP
7734.01)75.0( zP
2266.0)75.0( zP

The Symmetry Method????
Find Probability
that z > 0.75
Z-scores: -2 -1 0 1 2 3-3
0.75
Find area under the
curve to the right
of z = 0.75

Symmetry Method
Use symmetry of
the normal curve to
find area
Z-scores: -2 -1 0 1 2 3-3
0.75
Find area under the
curve to the right
of z = 0.75
- 0.75
2266.0)75.0( zP
2266.0)75.0( zP

Finding Area between two values
 Just use difference of the
two areas
az bz

az
Finding Area between two values
az bz bz
So,
)()()( abba zzPzzPzzzP 
)( bzzP 
)( azzP 
az

Difference of Area
Find Probability that
-1.25 < z < 0.75
Z-scores: -2 -1 0 1 2 3-3
0.75
Find area under the
curve between
z = -1.25 and 0.75
-1.25
)75.025.1(  zP
)25.1()75.0(  zPzP
1056.07734.0 
6678.0

Finding Probabilities of Normal Distributions
1. For data that is normally
distributed, find the percentage
of data items that are:
a) below z = 0.6
b) above z = –1.8
c) between z = –2 and –0.5
Always draw sketch, and shade
region!!!!

Finding Probabilities of Normal Distributions
2. Given a data set that is
normally distributed, find the
following probabilities:
a) P(0.32 ≤ z ≤ 3.18)
b) P(z ≥ 0.98)

Working with Normal Distributions
1. Don’t confuse z with x !!
Before solving real world applications of
data that is normally distributed, we need to
first calculate any appropriate z-scores based on
the data. This is called normalizing the data.
Recall…
2. Make sure the data is normally distributed



x
z

Systolic blood pressure readings are normally
distributed with a mean of 121 and a standard
deviation of 15. After converting each reading to
its z-score, find the percentage of people with the
following blood pressure readings:
a) below 142 )142( xP ?)( zP
z < 1.4
%92.919192.0 or

b) above 131 )131( xP ?)( zP
z > 0.67
%14.252514.0 or

c) between 142 and 154 )154142(  xP ?)(?  zP
1.4 < z < 2.2
%69.60669.0 or

The placement test for a college has scores that are
normally distributed with  = 500 and  = 100. If
the college accepts only the top 20% of examinees,
what is the cutoff score on the test for admission?
(hint: you’ll need to use the table first, and work back)
z > ????
20.0????)( zP
80.0????)( zP

Finding z-score from known probabilities
(or percentages)
39
845.0z80.0????)( zP

(hint: you’ll need to use the table first, and work back)
z > 0.845
20.0)845.0( zP
80.0)845.0( zP

z > 0.845 So, what is minimum test score?



x
z
100
500
845.0


x 5.584x

Demonstrating Importance of z - scores
 Lil’ Billy scores 60 on a vocabulary test and 80 on
a grammar test. The data items for both tests are
normally distributed. The vocabulary test has a
mean of 50 and a standard deviation of 5. The
grammar test has a mean of 72 and a standard
deviation of 6.
 On which test did the student perform better?
Why?

Demonstrating Importance of z - scores
 Lil’ Billy scores 60 on a vocabulary test and 80 on a grammar test.
The data items for both tests are normally distributed. The
vocabulary test has a mean of 50 and a standard deviation of 5.
The grammar test has a mean of 72 and a standard deviation of 6.
On which test did the student perform better? Explain why and
show all necessary work to support your conclusion.
Vocabulary (~Norm) Grammar (~Norm)
60vx 80gx
50v
5v
00.2vz
72g
6g
33.1vz

Classwork / Homework
• 10.4 Worksheet
• Page 638
•1 – 4, 9 – 19 odd, 25 – 35 odd

Lecture 10.4 bt

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Lecture 10.4 bt