Transcript standard deviation of 100

Honors Stats Day 14
Normal Model, 68-95-99.7
rule, Percents
Do Now
1) Sara scored an 87 on a quiz with a class average
of 84 and a spread of standard deviation 4.7.
• Gerardo scored a 90 on a quiz with a class average
of 87 and a spread of standard deviation 2.3.
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Who did better?
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2) Will ran a 6.2 in a race where the average race
time was 7.1 with a standard deviation of 0.6.
Melissa ran a 7.1 in a race where the avg time was
a 7.6 and a standard deviation of 0.3.
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Who is the more valuable teammate?
Try this…
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A highly selective university will only
admit students who place at least 2zcores above the mean on the ACT
that has a mean of 18 and a
standard deviation of 6. What is the
minimum score that an applicant
must obtain to be admitted to the
university?
Try another…
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A school is judging its soccer student
scholarships based on number of goals scored
in the season to incentivize the high scoring
forwards to attend the school.
It says that they will give a scholarship to
students who score more than 2 standard
deviations above the mean of 21.4 goals with
a standard deviation of 3.7. What does David
need to score in the season to get the
scholarship?
Review of Z-Score
Comparison
Below are the results from 3 athletes in a three-event track competition.
Decide who receives the gold medal based on the following results:
Hint: each event is a different set of data so we need to find z-score for
each event
Event
Competitor
100 m dash
Shot Put
Long Jump
A
10.1 sec
66’
26’
B
9.9 sec
60’
27’
C
10.3 sec
63’
27’3’’
Mean
10 sec
60’
26’
Standard
Deviation
0.2 sec
3’
6’’
What are we doing when
we make a z-score?
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Z-Score:
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a way to compare 2 different data sets with 2
different means
Step 1:
Step 2:
x – mean
÷ standard deviation
Example: data set where mean = 5,
sd = 2
Step 1:
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x – mean
Shifting all our data so that the mean
is 0
5
0
Step 2: divide
standard deviation
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This stretches the graph so that the scale is 1
0
-3 -2 -1 0
1
2
3
This is the STANDARD NORMAL MODEL.
It shows the distribution of z-scores!
A Normal Distribution
Standard Deviation
MEAN
Drawing Normal
Models
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Notation: N ( mean, standard dev )
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Example: N(18,3)
Sketch the Normal
Distribution
1) N(500, 100)
• 2) N(100, 16)
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Normal Model
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The Normal Model (z-scores) allow us
to:
Compare 2 different data sets (2
different means and sd)
2. Find the percentile or likelihood of an
event
1.
Knowing Extremes
Normal models show how LIKELY it is to find
a value that far from the mean
or what percentage of the data fell below,
above, or between given value(s)
Sketch and Describe
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Sketch the distribution of the
birthweights of babies: N(7.6lb, 1.3lb)
We are 68% sure that the baby will be born between
______lb and ______lb.
We are 99.7% sure that the baby will be born
between _____lb and _____lb.
What is the likelihood that the baby will be less than
8.9 lbs?
Side 1 of Practice
Sheet
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Complete the first side of the
practice sheet:
Using the 68-95-99.7 Rule
Example 1(on 68-95-99.7)
The SAT test has 3 parts:
Writing, Math, and Critical
Reading. Each part has a
distribution that is roughly
unimodal and symmetric,
an overall mean of 500
and a standard deviation
of 100 for all test takers.
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Suppose you earned a 600 on one part of the
SAT. Where do you stand among all students
who took the test?
1. Find Z score (600-500)/100 = 1
1. What percent is to the LEFT??
68% + ½(32%)= 84%
84th PERCENTILE
Percentiles
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What percentile are the following?
Example 2
(not on 68-95-99.7)
What if it is not EXACTLY 0, 1, 2, or 3
standard deviations away from mean?
• (what if z-score≠0,1,2,or 3)
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Suppose you earned a 680 on one part of the
SAT. Where do you stand among all students
who took the test? [N(500, 100)]
1. Find Z score (680-500)/100 = 1.8
1. What percent is to the LEFT??
WE HAVE A CHART FOR THAT
Z score chart
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What percentile is your score?
What percent above/below given number?
What percent chance or what is the likelihood of being
above/below/between given #s?
1) find z-score (if not given to you)
2) make picture (shade in what you want)
3) look in chart for percent on left
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Subtract from 100 if above/over- right side shaded
Example: On normally distributed graph N(15, 2), what
percent is above 13.7?
Example: z<-0.42
Second side of
Practice Sheet
Homework
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Ch. 6 #17, 18, 23, 29, 30