Download Chapter 13 part 2 - Department of Statistics

Reminders
• Email me and the TAs today if you have any
questions/concerns about grading of quiz 4 or HW 4
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Warm Up
• Final grades in BIO 180 at UW are Normally distributed
with a mean of 3.0 and a SD of 0.3. Final grades in
STAT 566 are Normally distributed with a mean of 3.7
and a SD of 0.1. If you get a 3.4 in BIO 180 and a 3.8 in
STAT 566, in which course is your standard score higher?
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Chapter 13 part 2: Normal calculations
Aaron Zimmerman
STAT 220 - Summer 2014
Department of Statistics
University of Washington - Seattle
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Percentiles
• The cth percentile of a distribution is the value such
that c percent of the observations lie below it and the
rest lie above it
? If you score at the 65th percentile on a test, then 65%
of the students who took the exam scored lower than you
and the other 35% scored higher
? If your height is at the 20th percentile within your
gender, then 20% of all people in your gender are shorter
than you and the rest are taller
Key idea #1: Each percentile in a normal distribution
corresponds with a standard score
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Percentiles
• Table B in the back of your book allows you to
“convert” between standard scores and percentiles
? For example, a standard score of -2.0 corresponds with
the 2.27th percentile (2.27% of observations in Normally
distributed data have a standard score less than -2.0)
? We’ve already seen this one, but the table will allow you
to do it for percents other than 68-95-99.7
Key idea #2: Percentiles can be interpreted as the area
under the standard normal curve to the left of the standard score
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How to use a Standard Score Table
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How to use a Standard Score Table
Check how the table works!
You can either:
• (1) Find a percentile given a score
? e.g. What percentile is associated with a score of -0.5?
• (2) Find a score given a percentile
? What standard score has 30% of standard scores less
than it?
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How to use a Standard Score Table
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ACT scores
• Remember that
ACT math scores
are normally
distributed with a
mean of 18 and a
SD of 6
• Grainne had a
standard score of
1.5 while Karthik
had a standard
score of -0.5 on the
ACT math test
• What were their
math scores?
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ACT scores
• The percentile of
Karthik’s score is
the area to the left
of his standard score
in the standard
normal distribution
? If you look up -0.5
in Table B, you see
that Karthik scored
at the 30.85th
percentile on the
ACT math test.
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ACT scores
• The percentile of
Grainne’s score is
the area to the left
of her standard
score in the
standard normal
distribution
? If you look up 1.5
in Table B, you see
that Grainne scored
at the 93.32th
percentile on the
ACT math test.
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What percentile corresponds to a standard score of -1.2?
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What percentile corresponds to a standard score of -1.2?
Answer: 11.51 percentile
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What standard score corresponds to the 90th percentile?
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What standard score corresponds to the 90th percentile?
Answer: Approximately a score of 1.3
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What percent of standard scores are between 0.5 and 1?
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What percent of standard scores are between 0.5 and 1?
Answer: 84.13 − 69.15 = 14.98 %
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What % of scores are larger than a standard score of 0.3?
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What % of scores are larger than a standard score of 0.3?
Answer: 1 − 61.79 = 38.21 %
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Strategy for problems involving percentiles of
Normal distributions
• Find the relevant mean
• Find the relevant standard deviation
• If you have an observation and want to calculate the
percentile (the percent below that value):
? Calculate the standard score
? Use Table B to convert to a percentile
? If you need the percent above the observation, calculate
the percentile and subtract from 100
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Distribution of heights
The heights of men are Normally distributed with a mean of
70 in. and a SD of 2.5 in. The heights of women are Normally
distributed with a mean of 66 in. and a SD of 3 in.
• If Karthik is 75 in. tall,
what percentile is his
height?
? Relevant mean: 70 in.
? Relevant SD: 2.5 in.
? Standard score:
75−70
=2
2.5
• Percentile: 97.73
• Conclusion: 97.73% of
men are shorter than
Karthik
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Distribution of heights
The heights of men are Normally distributed with a mean of
70 in. and a SD of 2.5 in. The heights of women are Normally
distributed with a mean of 66 in. and a SD of 3 in.
• If my little sister is 58.5
in. tall, what percent of
women are taller than her?
? Relevant mean: 66 in.
? Relevant SD: 3.0 in.
? Standard score:
58.5−66
= −2.5
3.0
• Percentile: 0.62
• Conclusion: 1-0.62 =
99.38% of women are
taller than Eliana
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Heights of maple
trees are Normally
distributed with a
mean of 34 ft. and a
standard deviation of
9 ft., while the
heights of elm trees
are Normally
distributed with a
mean of 19 ft. and a
standard deviation of
3 ft.
What percentile is
an elm tree that is
20.5 ft. tall?
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Heights of maple
trees are Normally
distributed with a
mean of 34 ft. and a
standard deviation of
9 ft., while the
heights of elm trees
are Normally
distributed with a
mean of 19 ft. and a
standard deviation of
3 ft.
What percent of
maple trees are
taller than 25 ft.?
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Strategy for problems involving percentiles of
Normal distributions
• Find the relevant mean
• Find the relevant standard deviation
• If you have an observation and want to calculate the
percentile (the percent below that value):
? Calculate the standard score
? Use Table B to convert to a percentile
? If you need the percent above the observation, calculate
the percentile and subtract from 100
• If you have a percentile and need the observation value:
? Find the standard score in Table B for that percentile
? Work backwards:
observation = (standard score) × SD + mean
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Distribution of heights
The heights of men are Normally distributed with a mean of 70 in. and a
SD of 2.5 in. The heights of women are Normally distributed with a
mean of 66 in. and a SD of 3 in.
• If you are a woman, how tall
must you be to be at the 90th
percentile?
? Relevant mean: 66 in.
? Relevant SD: 3.0 in.
? Percentile: 90
? Standard score: about 1.3
• obs = (1.3) × 3 + 66 = 69.9
• Conclusion: A woman must be
69.9 in. tall to be at the 90th
percentile
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Distribution of heights
The heights of men are Normally distributed with a mean of 70 in. and a
SD of 2.5 in. The heights of women are Normally distributed with a
mean of 66 in. and a SD of 3 in.
• If you are a man, how tall
must you be to be in the top
35% of men’s heights?
? Relevant mean: 70 in.
? Relevant SD: 2.5 in.
? Percentile: 65
? Standard score: about 0.4
• obs = (0.4) × 2.5 + 70 = 71
• Conclusion: A man must be 71
in. tall to be in the top 35% of
men’s heights
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Heights of maple
trees are Normally
distributed with a
mean of 34 ft. and a
standard deviation of
9 ft., while the
heights of elm trees
are Normally
distributed with a
mean of 19 ft. and a
standard deviation of
3 ft.
How tall is an elm
tree at the 10th
percentile?
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Heights of maple
trees are Normally
distributed with a
mean of 34 ft. and a
standard deviation of
9 ft., while the
heights of elm trees
are Normally
distributed with a
mean of 19 ft. and a
standard deviation of
3 ft.
How tall does a
maple tree need to
be to be in the top
1 % of all maple
trees?
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Homework #5
• Finish reading Chapter 13 if you haven’t already
• Do problems 13.21, 13.22, 13.23, 13.24, 13.25, 13.27,
13.29, 13.30
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