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Stats Day 22
More on Z-Scores
SILENT DO NOW
DO NOW:
ACT Half Sheet
HW due TOMORROW:
Ch. 6 #17-19, 29, 33, 34
Objective
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SWBAT calculate
likelihood/probability using a z-score
chart
SWBAT calculate a missing value
given its likelihood/percentile and the
normal distribution (mean, sd)
Practice 68-95-99.7
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On the Boards!
Example 2
(not on 68-95-99.7)
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What if it is not EXACTLY 0, ±1, ±2, or
±3 standard deviations away from
mean, how do we calculate
probability?
This is where we use the
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STANDARD NORMAL MODEL
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The Normal
Distribution
Notice how each line
represents the
number of standard
deviations away
from the mean..or
the…
Z-SCORE!!!
Standard Deviation
MEAN
The Standard Normal
Curve
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The Distribution of Z-SCORES!
THERE ARE NO UNITS! It’s the
same for every set of data, no
matter what you are talking
about!
Z-score as a shift and
rescale of distribution
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When we make a z score, we take each data
point and subtract a number (the mean)
What would happen to the graph y=x if we subtracted 1 from each x
value y=(x-1)?
What changes?? We shift everything over (mean becomes 0)
We subtract EVERY VALUE by that number
What does not change??
SPREAD: Standard Deviation, Range, IQR
Shift and Rescale
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When we make a z score, we also take
each data point and divide by a number
(the standard deviation)
CHANGING OUR SCALE TO
STANDARD DEVIATION
The Standard Normal
Curve
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The Distribution of Z-SCORES!
THERE ARE NO UNITS! It’s the
same for every set of data, no
matter what you are talking
about! YAY!
How can this help
us?
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N(500,100)
Suppose you earned a 600 on one part of the
SAT. What percentile are you?
1. Find Z score
2. What percent is to the LEFT??
We also have a chart!
Since there are no units, we can use it for
everything!
Yesterday…
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Focused on Normal Distribution, but
saw that we could shift the normal
distribution to show a distribution of zscores: the Standard Normal Curve
• Today…
• We will use z-scores the Standard
Normal Curve Chart to determine
percent and probability
If it’s not exactly ±1,
±2, or ±3 SDs from
the mean
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If your z-score is not exactly ±1, ±2, or ±3,
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The chart can tell you the percent!
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N(500,100)
Suppose you earned a 680 on one part of the
SAT. What percentile are you?
1. Find Z score
2. What percent is to the LEFT??
THE CHART ONLY GIVES YOU
WHAT IS ON THE LEFT!
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N(500,100)
Suppose you earned a 720 on one part of the
SAT. What percent of test takers did better
than you?
1. Find Z score
2. What percent is to the RIGHT??
(Draw a picture)
The percent the chart will give you is
the percent to the left…
To get the right, subtract from 100!
Z-SCORE CHART
1.
2.
3.
4.
Find your z-score
Draw a picture- vertical line at zscore and shade
Look up z-score to tell you percent to
the LEFT of your score (less than your
score) Note: this is the percentile
Determine if you want that percent
(less than) or 100-[that percent]
(more than)
Z-Score
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1. What is the likelihood that you
scored below an 18on this last ACT
given N(18.8, 3.7)?
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2. That you scored above a 20?
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3. That you scored above a 22.5?
Find the following in
pairs
1) z < 2.88
• 2) z<-0.11
• 3) z > 0.24
• 4) -1.31 < z
• 5) Drink prices at a coffee shop are
distributed normally N(3.25, 1.00).
What is the probability that your
drink will cost more than $5?
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Challenge
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Life span is distributed normally
N(71, 14). What is the probability
that you will live to at least 75 but no
older than 90?
Hint: Draw a Picture!
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Find the percentages for each and
subtract the larger from the
smaller!
Finding the Between
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Find the percentages for each and
subtract the larger from the smaller!
Practice again!
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What is the chance that a baby
weighs between 4.5 oz and 8.5 oz
given N(6, 1.7)?
Stats Day 23
Final Z-Score Game
SILENT DO NOW
ON DESK:
Chapter 6 #17-19, 29, 33, 34
DO NOW:
Check HW with Grading Pen
Homework: Bring Book on Monday
Due TUESDAY
Ch. 6 #31, 32, 36, 38, 41, 43
**Ch 6 Quiz Tuesday
Quick Review
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**DRAW PICTURE
P3 Class grades are normally distributed with
a mean of 83 and a standard deviation of 5.1.
What is the probability that a new kid
entering my class will…
• 1) get below an 80%?
• 2) Above a 92%?
• 3) Between an 85 and a 95%?
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Trashketball!
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You can use your notes
Everyone in your group must have
an answer
To get the shots, everyone in group
raises hands with completed work
raised hand and not done?
Disqualified from round
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After each problem, the winning
group will show work
Question 1
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What percentile am I if I scored a 25
on the ACT that is normally
distributed with N(21, 3.2)?
Question 2
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What is the chance that I scored
above a 93 on a test that was
normally distributed as follows:
N(88, 3.6)?
Question 3
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What percent of basketball players
have less than a .6 shooting
percentage at the free-throw line if it
is normally distributed N(.71, .09)
Question 4
What percent of students score
between a 21 and a 26 on the ACT
that is normally distributed
• N(21, 3.2)?
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Question 5
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What percent of M&M packages
weigh between 2.4 and 2.6 ozs given
N(2.43, 0.4)?
Question 6
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What percent chance is there that
you will score more than 2 goals in a
game given N(2.4, 0.8)
Question 7
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Chip bag weights are normally
distributed with a mean of 1.8 oz
with a SD of .1oz. What is the
probability of getting a chip bag that
weighs 2 oz or more?
Question 8
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Labrador retriever life spans are
normally distributed N(11.8, 1.3).
What is the probability that your
chocolate lab lives between 14 and
15 years?
Question 9
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The average room temperature in
Pritzker is 70 degrees Fahrenheit
with an SD of 5 degrees. What is the
probability that your next room will
be between 68 and 72 degrees?
Exit Ticket
Finding scores from
percents
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Say a school only accepts students in
the top 10%(90th percentile). If the
ACT is normally distributed N(21,
3.2), what score would you need to
get on your ACT?
WORK BACKWARDS
1. Draw a picture
2. Determine percent to the left
3. Find z in chart from percent
4. Plug in equation and solve for x
Another Example
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The organic chemistry class at
Boston College, after their first test,
suggests to its lowest 15% of
students that they should drop the
course. If the class has a distribution
of N(78, 8.2), what score would you
have to be above to stay in the
class?
Another Example
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My friend just had a premature baby
that was born in the 30th percentile
of baby weights. If newborns have a
normally distributed weight modeled
by N(7.5lbs, 1.25lbs), how much did
my friend’s baby weigh?
Stats Day 24
Ch 6 Review
SILENT DO NOW
DO NOW:
ACT Halfsheet
Homework:
Due TOMORROW
Ch. 6 #31, 32, 36, 38, 41, 43
**Ch 6 Quiz TOMORROW
Chapter 6 Topics
1.
2.
3.
4.
5.
6.
Definition and formula of Z-Score
Z-Score to compare two different data sets
Normal Curve and 68-95-99.7 rule
Standard Normal Curve
Definition of percentile
Z-score to find probability/percentile
Less than
2. Greater than
3. Between
1.
7.
8.
Given probability/percentile, find x
Any relevant notation
Sample Study Guide
In Class Study Guide
and Practice Quiz
Stats Day 25
Ch 6 QUIZ
Finding z-score on
calculator!
Unit 1 Group Teach
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Each group will re-teach objectives to the class
You must teach objectives with key points and
at least 2 examples
You will have max 10 min to teach
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Begin prep with your group NOW!
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Homework
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Unit 1 Review Packet DUE in 3 DAYS
This will be “Schrishuhn Style”
You must complete the entire packet.
I will assign a problem to students randomly and all
problems will be put on the board.
When you finish, check with me if it is correct. If you
are correct you may do a room walk/visit others to
check the rest of your answers and ask questions. (I
suggest doing corrections in another color).
Remember, this is studying for your test!
When you have checked and visited you may sit down
and try some extra credit!
Stats Day 22
Group Teach
SILENT DO NOW
ON DESK:
Group Teach Notes from Yesterday
DO NOW:
Spend 10 minutes preparing with
partner for Group Teach
Group Teach
Homework
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Unit 1 Review Packet DUE in 2 DAYS
This will be “Schrishuhn Style”
You must complete the entire packet.
I will assign a problem to students randomly and all
problems will be put on the board.
When you finish, check with me if it is correct. If you
are correct you may do a room walk/visit others to
check the rest of your answers and ask questions. (I
suggest doing corrections in another color).
Remember, this is studying for your test!
When you have checked and visited you may sit down
and try some extra credit!
Stats Day 23
Group Teach
SILENT DO NOW
ON DESK:
Group Teach Notes from Yesterday
DO NOW:
Spend 10 minutes preparing with
partner for Group Teach
Group Teach
Homework
•
Unit 1 Review Packet DUE TOMORROW
This will be “Schrishuhn Style”
• You must complete the entire packet.
• I will assign a problem to students randomly and all
problems will be put on the board.
• When you finish, check with me if it is correct. If you
are correct you may do a room walk/visit others to
check the rest of your answers and ask questions. (I
suggest doing corrections in another color). Remember,
this is studying for your test!
• When you have checked and visited you may sit down
and try some extra credit!
• We will go over any further questions in the end!
•