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Stat3.1Lessonplan(3days)
•
•
1) SWDAT define and calculate percentiles, z-score, standardized
value, and center and spread of density curves.
– CC.9-12.S.ID-4
• 2) Anticipatory set: Students will be asked what it means for
their test score to be in the 96th percentile.
– Procedure: Students will formulate a definition for percentile
and then calculate percentiles for various data points and data
sets. Then, students will calculate z-scores and learn about
standardizing. Lastly, I will introduce density curves, their
properties, and uses in statistics along with various exercises.
– Closure: Review strategy for exploring data
– Resources: Slides, Textbook,
– Adaptations: Work with individual students or small groups as
needed. Demonstrate activities as needed.
– Homework: Various book exercises
3) Assessment: Informal during group and individual work. Select
students to share and justify their solutions and explain strategies
for various problems.
Section3.1Donow:
• Takesomeshellnotes
• Whatdoesitmeanwhenyouaretoldthatyou
areinthe96th percentileonanexam?
• The57th percentile?
Percentiles
• ThePth percentileofadistributionisthe
valuewithppercentoftheobservationsless
thanorequaltoit?
Relatethisdefinitionbacktoyouranswerforthedo
now.Wasyourmeaningcorrect?
Let’slookatthe65-inchheight.Whatisthe
percentileoftheheight?
Since10ofthe25observationsareatorbelow
thisheight,65-inchesisthe40th percentile.
Standardizing
• Wecanalsodescribelocationsbycalculating
thenumberofstandarddeviationsan
observationsfallsaboveorbelowthemean.
• Convertingobservationsfromoriginalvalues
tostandarddeviationsiscalledstandardizing
• Thenewstandard-valuesarecalledz-scores.
Findthestandardizedheightorz-scorefor65
inchesand74inches.
Z=(65-67)/4.29=-.47sothisheightisahalfofa
standarddeviationawayfromthemean.
Z=(74-67)/4.29=1.63
A)Findthepercentilecorrespondingto6.35million.
Whatdoesthisvaluemean?
B)Findthez-scorecorrespondingto6.35million.
Whatdoesthisvaluemean?
DensityCurves
• Exploringdataonasinglequantitativevariable
• 1.Alwaysplotyourdata:makeagraph,usuallya
histogramorstemplot
• 2.Lookfortheoverallpattern(shape,spread,
center)andforoutliers
• 3.Chooseeitherthe5-numbersummaryormean
andstandarddeviation
• 4.Sometimestheoverallpatternofalarge
numberofobservationsissoregularthatwecan
describeitbyasmoothcurve.
Histogramsandsmoothcurves
-Curvetoshowtheproportionof
observationsinanyregionby
areasunderthecurve.
-Thetotalareaunderthecurveis
alwaysmadetobe1.
(a) The proportion
of scores less
than 6.0 from the
histogram is
0.303.
(b) The proportion
of scores less
than 6.0 from the
density curve is
0.293.
Densitycurves:CenterandSpread
• Median- Thepointonthecurvewherethe
areasoneachsideareequal.
• Mean- Thepointatwhichthecurvewould
balance
• Q1andQ3arerepresentedbytheareasunder
thecurvethatareone-quarterofthewayand
three-quartersofthearea.
The mean of a density curve
is the point at which it
would balance.
The median and mean for two density curves:
a symmetric Normal curve and a curve that
is skewed to the right.
Classwork:pg.111#
3.9-3.15
0.5
3.9
2
Median = 1; 0.75 =
2
Q3 = 1.5.
Answers will vary. One possibility:
The distribution is skewed to the right so that the median < mean.
median is approximately 3.5 and the mean is approximately 4.2.
The
3.10 Answers will vary. One possibility:
The distribution is well approximated by a uniform distribution between 0 and
9 having mean=median=4.5.
3.11
(a)
The overall shape of the distribution is symmetric and mound-shaped.
Hence, the mean and median are equal and are at point A.
(b)
The overall shape of the distribution is skewed to the left. The mean is at A
and the median is at B.
3.12
and the median is at B.
3.12
3.13
(a) The distribution is skewed to the right, thus the mean is to the right of the
3.13 median. Point “A” clearly does not mark the 50th percentile, so the median
(a) must
The distribution
is skewed
to at
the“C”.
right, thus the mean is to the right of the
be at “B” and
the mean
median. Point “A” clearly does not mark the 50th percentile, so the median
(b) The
distribution
is the
symmetric
so
the mean and the median both mark the
must
be
at
“B”
and
mean
at
“C”.
th
distribution
located
at “A”.
50
(b) The percentile
distributionofisthe
symmetric
so and
the are
mean
and the
median both mark the
(c) The
distributionof is
to the
theatmean
theskewed
distribution
andleft,
are thus
located
“A”. is to the left of the
50th percentile
th
(c) median.
The distribution
is skewed
to thenot
left,
thusthe
the50mean
is to the
the
percentile
so left
the of
median
Point “C”
clearly does
mark
median.
“C” the
clearly
does
not mark the 50th percentile so the median
must
be atPoint
“B” and
mean
at “A”.
must be at “B” and the mean at “A”.
3.14
means that
that 90%
90% of
of similar
similar men
men have
have cholesterol
cholesterollevels
levelsthat
thatare
are
3.14 It
It actually
actually means
less
or equal
equal to
to Martin’s
Martin’s level.
level. Martin’s
Martin’s cholesterol
cholesterolisiscomparatively
comparatively
less than
than or
high, not low.
3.15
(a)
(b)
(c)
The mean of the 50 observations is calculated to be 5.598 percen
median is 5.45 percent. The standard deviation is 1.391 percent
distribution of unemployment rates is fairly symmetric, with a center o
5.5 percent and a spread of 5.6 percent.
In the ordered data set, Illinois’s unemployment rate is 40th. Th
percentile for Illinois is 40/50=.8. 80% of the unemployment rates a
than or equal to Illinois’s unemployment rate. Illinois has a pret
(b)
(c)
(d)
The mean of the 50 observations is calculated to be 5.598 percent. The
median is 5.45 percent. The standard deviation is 1.391 percent. The
distribution of unemployment rates is fairly symmetric, with a center of about
5.5 percent and a spread of 5.6 percent.
In the ordered data set, Illinois’s unemployment rate is 40th. Thus, the
percentile for Illinois is 40/50=.8. 80% of the unemployment rates are less
than or equal to Illinois’s unemployment rate. Illinois has a pretty high
unemployment rate compared to the remaining 49 states.
Since .4*50=20, the state whose unemployment rate is 20th among the
ordered observations represents the 40th percentile. Texas represents the
40th percentile with an unemployment rate of 5.1%. The z-score for this
5.1 5.598
state is z
1.391
0.36.
3.16
(a)
The unemployment rate for Illinois was 2.4% (6.9-4.5) higher in September,
2008 than in December, 2000.
(b)
The percentile for Illinois (its unemployment rate compared to the rest of the
country) did not change much from December of 2000 to September of
2008, it actually dropped slightly. It was at the 86th percentile in December
of 2000 (43/50=0.86) versus the 80th percentile in September of 2008.
(c)
The z-score for Illinois in December of 2000 was z
4.5 3.47
1.03. The
1
unemployment rate for Illinois was slightly higher in December of 2000 than