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Stats Day 21
More on Z-Scores and the Normal
Model (Using the Chart)
SILENT DO NOW
ON DESK:
Chapter 6 #5-10
DO NOW:
Check HW with Key and Grading Pen
ACT Half Sheet
Homework DUE FRIDAY:
Chapter 6 #17-19, 29, 33, 34
Objective
•
SWBAT use the z-score and z-score
chart to identify percentiles and
likelihood
Working with 68-9599.7 Rule
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•
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A lightbulb on average lasts 500 hours with σ=30
1) What percent of lightbulbs last between 440
and 560 hours?
2) What percent of lightbulbs last over 590
hours?
3) What percent of lightbulbs last less than 470
hours?
4) If you have a lightbulb that says it lasts 530
hours, what percentile is it?
Example 1(on 68-95-99.7)
The SAT test as 3 parts:
Writing, Mth, 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. Draw Normal Curve
2. What percent is to the LEFT??
68% + ½(32%)= 84%
84th PERCENTILE
Percentiles
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What percentile are the following?
(percent less than given number)
Challenge: Find the
number that cuts off
the percentiles
N(35, 5)
84th percentile:
97.5th percentile:
50th percentile:
Worksheet
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Z-score empirical rule ws
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?
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!
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!
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 above a 20 on this last ACT
given N(18.8, 3.7)
2. That you scored below an 18?
Find the following
1)
• 2)
• 3)
• 4)
• 5)
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z < 2.88
z<-0.11
z > 0.24
-1.31 < z
0.12 < z
Exit Slip!