| SPSS #A due today, rest due this coming Monday. Hand in: Bring also exam questions! Normal distribution: (Normal templates-optional-you can count squares--may help!) p. 74 3.5 Women's hts, sketch p. 74 3.6 Normal, women's hts--68-95-99.7 rule. p. 74 3.7 pregnancies--68etc rule (This distribution may not apply to planned births, of which we now know there are a lot!) p. 88, 3.51 check 68-95-99.7 rule , using applet: Normal Density Curve on your CD or at http://bcs.whfreeman.com/bps4e. - - - - - Standardize: Draw and label the normal density curve, the "raw" axis and the "z" axis together, mark your value(s), as well as calculating. p. 76, 3.9 mens & women's heights p. 86, 3.33 ACT/SAT Jacob and Emily (Info is above #3.32) --Exam covers to here-- -- Postpone ALL the rest (but you can start them using the Applet!)-- - Using table with "z"'s--standard normal.--------- Table use--z: Always sketch a normal curve first, mark the area you are looking for! Do these with the Applet: Normal Density Curve on your CD or at http://bcs.whfreeman.com/bps4e. , and check with your table answers. (Uncheck the 2-tail box for most uses. Mean 0, s.d. 1) p.80 3.10 z's to proportions, using Table A. - -- "Backward"--z :Always sketch a normal curve first, roughly mark the proportion=area you are given. p.83, 3.13 (backward z) Do with table, check using Applet: Normal Density Curve on your CD or at http://bcs.whfreeman.com/bps4e. p. 89, 3.52 Quartiles of normal dist. Use the Applet and also, use table A to find the quartiles. Your answers may differ in the second decimal place because the Applet only goes by .02's on the z-axis --.64, .66, .68... and Table A goes by .01's. Start the following, using the APPLET--keeping your paper to complete with the next assignment(s). Do hand computations as we learn how!. = = = = Using table with "x"'s--"raw" values. = = = = Begin these by drawing and labeling the appropriate normal curve for each question, leaving space for computation. Normal templates-may help. Then use the Applet: Normal Density Curve on your CD or at http://bcs.whfreeman.com/bps4e. to find the required values. Write these on your paper. Next, calculate the values using Table A. Your answers from each method should be very close (the Table gives a bit more accuracy than the Applet.) p. 87, 3.37 Jacob's score, and 3.39 top score. Mean, s.d. are before 3.32 on p. 86. p. 87, 3.46 surprising difference in tails A. , What proportion of pregnancies last 310 days or more? Find Mean and s.d. in p.74, 3.7 (more next time on this) p. 80-81 3.11 and 3.12 (locomotive adhesion, 2 dist's) ..------- "Backward Normal"----------- Begin these by drawing and labeling the appropriate normal curve for each question, leaving space for computation. Normal templates-may help. Then use the Applet: Normal Density Curve on your CD or at http://bcs.whfreeman.com/bps4e. to find the required values. Write these on your paper. Next, calculate the values using Table A. Your answers from each method should be very close (the Table gives a bit more accuracy than the Applet.) p. 83, 3.14 IQ test p. 87, 3.41 Abigail, top 20%. Mean, s.d. are before 3.32 on p. 86. p. 87, 3.42 quartiles Mean, s.d. are before 3.32 on p. 86. p. 179, 7.27 breaking bolts, (a, b +). For (a), think carefully about which side of 90 you want: Does a bolt that breaks at 95 ksi qualify? Does a bolt that breaks at 85 ksi qualify? ALSO: If they test every bolt and throw away all bolts that break at 70 ksi or below, what proportion do they throw away? |
Read, to discuss | Optional
(more practice) ---------------- Postpone the rest p. 86 3.30 z's to proportions "Backward" - - - - - - - - - - - - - - - "raw", "backward" p. 87, 3.43, quintiles Mean, s.d. are before 3.32 on p. 86. Quintiles are used by the government to report much economic census data. |
Continue here Wednesday, after
questions for Exam.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Standard normal table use~ ~ ~ ~ ~ ~ ~
~ ~ ~
Standard Normal table use. Our tables give area to the left of a z value (Cumulative Proportions)
Using standard normal table: See text p. 76-80. Table A: p.684-5. Table A
(Excel)
z | .00
.01 .02 ..... =number
in "hundredths place"
...|
-2.4 | .0082 .0080 .0078 ....=
area to the left of "edge number"
...|
1.4 | .9192 .9207 .9222
ones&tenths
Proportion of z's below -2.40
= P(z < -2.40) = .0082
= prop. of individuals 2.40 s.d.'s or more below the mean)
P(z
< -2.41)
= .0080 P(z < -2.42) = .0078 ,
P(z < 1.42) = .922
?z has more than 2 dec. places? Round to 2.
Sketch the density, mark the area
you're looking for.
Figure out how to get it using areas to
the left of one or more z-values.
Think cutting up paper
bell-curves.
(Remember whole area is 1.)
Example: Proportion of observations between 0.5 and
1.4
P(0.5 < z <1.4) =
Proportion of observations below 1.4 minus
Proportion
of observations below 0.5
P (z < 1.4) - P(z < 0.5) = .9192 - .6915 =
.2277
.
Reading table backward: Table A
(Excel)
What z value has area ..... to the left/right
of it?
Sketch roughly.
Restate
(if
needed) as "What z value has area A to the LEFT of it."
Look
in body of table for the value closest to A.
Go
to edge(s) of table to find what z that goes with.
Example: "What z value has 10%
of the observations above it?" This is the same z as the one for:
"What
z value has 90% of the observations below (to the left of) it?"
"What z value is at the 90th percentile?"
Find
in the table .8997 and .9015 -- .9000, our number, is
between them.
.8997 is a little closer to.9000, so use it.
For .8997, the z value is 1.28.
1.28 has 10% of the observations above it.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - - -
All of these can be checked using the Applet:
Normal Density Curve http://bcs.whfreeman.com/bps4e
Standardizing: A way of
comparing an
individual
against
its pack.
Comparing individuals from different packs, each relative to its own.
Removes "units of measurement" from the discussion.
Enables use of the standard normal table.
Examples: ("Classic IQ test", mean 110, s.d. 25)
85 is 1 s.d. below
the mean. Computation: z = (85 –
110)/25
= (–25 raw points)/25
= –1 s.d. from mean.
145
is
how many s.d.'s above the mean?
Computation: z = (145
– 110)/
25=
(35 raw points above mean)/25
=
1
2/5 = 1.4 s.d. above mean
- - - - - - - - - - - - - - - - - - - - - - -
-
- - - - - - - - - - - - - - - - - - - -
"What proportion"problems: BPS4e pp. 78-80
or P ( 100 < x <
145)
= P ( –.4 < z < 1.4) = P( z < 1.4) – P(z < –.4) = .9192 –
.3446
= .5746
Read
"Proportion
of x's with 100 <x<145" for P(100<x<145)
- - - - - - - - - - - - - - - - - - - - -
- - - - - - - - - - - - - - - - - - - -
"Backward problems" "What
raw (x) value has area ___ to the
left/right
of it?"
BPS4e pp. 81-83.
Sketch the curve, labeled with x values and z values, and the
Area, roughly.
Restate
(if
needed) as "What z value has area A to the LEFT of it."
Look
in body of table for the value closest to A.
Go
to edge(s) of table to find what z that goes with.
Convert
the z to an x: z is
the number
of standard deviations above the mean.
Multiply z by the size of 1
standard
deviation. Now you have distance
above
the mean, measured in raw units.
Add the mean.
Now you have the "raw" value x.
(You have "unstandardized")
Example: What x
value
has 10% of the observations above it? This is the same
x as the one for:
What
x value has 90% of the observations
below
(to the left of) it.
The table gives z = 1.28, approximately. Table
A (Excel)
The "Classic IQ test"score x=
mean
+ z (s.d.) = 110
+ 1.28 (25)= 110
+ 32 = 142
Percentiles: a "Classic IQ test" score of 142 has 90% of the scores at or below it. 142 is the 90th percentile.
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