Download Normal Distributions

Probability and Statistics
Name_______________________________
Unit IV: Randomness in Data
Topic 15: Normal Distributions
Worth 90 Points
(In-Class Activities pages worth 5 points each / Homework Activities pages worth 10 points each)
Normal Distributions
Topic 15:
Preliminaries
(page 325)
(page 326)
1.
I believe a typical baby weighs
pounds at birth.
2.
I would guess that
% of babies would weigh less than 6 pounds.
I would guess that
% of babies would weigh more than 10 pounds.
3.
I would expect Professor
to assign more A’s.
4.
I would expect Professor
to assign more F’s.
5.
I would prefer to take a course with Professor
factors are equal.
6.
I believe that a human pregnancy lasts on the average
7.
I would guess that
% of human pregnancies last less than 8 months.
I would guess that
% of human pregnancies last more than 10 months.
8.
I believe Forrest Gump’s IQ was
, if all other
.
days.
Activity 15-1:
(a)
Placement Scores and Hospital Births
(page 327)
The similarities that I notice about the shapes of these distributions is that they are …
.
(b)
[Draw a sketch of a smooth curve that approximates the general shape of the histograms.]
Data that display the general shape shown above occur frequently. Theoretical mathematical
models used to approximate such distributions are called
.
Each normal distribution shares 3 characteristics:
(1)
Every normal distribution is
.
(2)
Every normal distribution has a single peak at its
(3)
Every normal distribution follows a
.
-
curve.
Each normal distribution is distinguished (identified) by 2 things:
(1)
its
(2)
its
The mean
and
.
determines where its center is; the
at its mean and its point of
The standard deviation
The distance between the mean
to the
of a normal curve is
.
indicates how spread out the distribution is.
and the points where the curvature changes is equal
.
PUT GRAPH HERE!
(c)
[Label each curve with A, B, or C.]
PUT GRAPH HERE!
COPY OF TABLE FROM PAGE 328!
(d)
Are these proportions quite close to each other and to the
predictions of the empirical rule that you studied in Topic 5?
z-score =
(e)
(f)
observation - mean ____! ____
=
standard deviation
z =
=
(show work)
(answer rounded to nearest hundredth)
(show work)
(answer rounded to nearest hundredth)
z =
=
(g)
The thing true about these 2 scores is that they are
(h)
The interpretation of these z-scores is that the 2 values fall
deviation below their respective
.
.
of one standard
(i)
The proportion of observations falling below 8.5 in the placement exam data is
%.
(j)
The proportion of observations falling below 23.5 in the simulated births data is
%.
Are these proportions fairly close?
The closeness of these percentages indicates that to find the proportion of data falling in a
given region for normal distributions, all we need to determine is the
.
Values with the same z-score will have the same
of observations
lying below them, for any normal distribution. Thus,
percentages for
of finding
normal distributions, we need only the percentages corresponding
to these z-scores.
(k)
The probability that the randomly selected student’s score will be below 8.5 is
%.
The probability of a randomly selected observation falling in a certain
is equivalent to the proportion of the population’s observations falling in that interval.
Since the total area
the curve of a normal distribution is
, this
probability can be calculated by finding the area under the normal curve for that interval.
The total area
(l)
the curve is always equal to
or
Area corresponding to the probability of a placement score falling below 8.5 =
(m) Is this value reasonably close to your answers for the proportions
of the placement and birth data less than their z-scores of -0.45?
% .
.
Use
to denote the standard normal distribution.
The notation Pr ( a < Z < b ) denotes the probability lying
a and b, calculated as the area
the values
the standard normal curve in that region.
The notation Pr ( Z ≤ z ) denotes the area to the
Pr( Z ≥ z ) refers to the area
of a particular value z, while
a particular z value.
EXAMPLES
NOTE:
(1)
Sketch the curve and the area under the curve, then find the proportion.
Pr (Z < 0.68) =
=
(3)
(2)
%
Pr (Z > 0.68) =
=
=
(4)
%
Pr (Z ≤ 0.68) =
%
Pr (Z < -1.38) =
=
%
(5)
Pr (Z > - 1.75) =
(6)
=
(7)
%
=
Pr (-1.38 < Z < 0.68) =
=
(9)
Pr (Z < k) = .8997
k=
Pr (Z < -2.49) =
(8)
Pr (Z < -3.81) is
(10)
Pr (Z > k) = .0630
%
k=
%
Birth Weights
Activity 15-2:
(pages 331 to 334)
mean µ = 3250 grams and standard deviation ó = 550 grams
(a)
(b)
My guess as to the proportion of babies born with a low birth weight is
The technique of
allows you to use simple algebra to convert
any normal distribution into a
normal distribution.
If X denotes any normal distribution, then standardization says that, Z =
(c)
z-score =
=
(show work)
) =
(answer rounded to nearest hundredth)
(d)
Pr ( Z <
(e)
[Follow directions with the calculator.]
=
%
Does your answer match (to within rounding discrepancies) your answer to (d)?
(f)
[Use Standard Normal Probabilities Table.]
z-score =
=
(show work)
Pr ( Z >
) =
(answer rounded to nearest hundredth)
=
%
[Use your calculator.]
Pr ( X > 4536) =
(g)
% .
[Know both ways in (f).]
=
%
.
(h)
[Use Standard Normal Probabilities Table.]
z-score =
=
z-score =
=
Pr (
<Z<
) =
-
=
%
[Use your calculator.]
Pr ( 3000 < X < 4000 ) =
(i)
Proportion for low birth weight babies is
=
%
%.
Proportion for babies weighing between 3000 and 4000 is
%.
[comment]
(j)
A newborn would have to weigh
grams to be among the lightest 2.5%.
(k)
A newborn would have to weigh
grams to be among the heaviest 10%.
Activity 15-3:
Matching Samples to Density Curves
While normal distributions are the
common, they are not the only kind of
theoretical probability model. Any curve under which the total area is
which areas correspond to
(c)
Sample 1: Population
and for
represents a probability model. Such
curves are called
(a)
(pages 334 to 336)
.
(b)
Sample 5: Population
Sample 2: Population
Sample 6: Population
Sample 3: Population
Sample 7: Population
Sample 4: Population
Sample 8: Population
It is easier to discern the shape of the population from a sample size of
Especially with
not look very
.
sample sizes, sample data from normal populations may
and may be hard to distinguish from sample data
from other
of populations.
Activity 15-4:
Normal Curves
a:
mean µ =
and standard deviation ó =
b:
mean µ =
and standard deviation ó =
c:
mean µ =
and standard deviation ó =
d:
mean µ =
and standard deviation ó =
(pages 337 & 338)
Pregnancy Durations
Activity 15-5:
(page 338)
mean µ = 270 and standard deviation ó = 15
(a)
The proportion of all pregnancies that last less than 244 days is
% .
(approx. 8 months)
[Show work here.]
(b)
[Sketch curve here.]
The proportion of all pregnancies that last more than 275 days is
% .
(approx. 9 months)
[Show work here.]
(c)
The proportion of all pregnancies that last over 300 days is
[Show work here.]
(d)
[compare]
% .
[Sketch curve here.]
The proportion of all pregnancies that last between 260 and 280 days is
[Show work here.]
(e)
[Sketch curve here.]
[Sketch curve here.]
% .
Professors’ Grades
Activity 15-6:
(a)
(b)
(pages 338 & 339)
[Sketch both of the teachers’ grade distributions on the same scale]
|
|
|
|
|
|
|
|
|
|
0
10
20
30
40
50
60
70
80
90
|
100
The professor that gives the higher proportion of A’s is Professor
.
[Show the appropriate calculations to support your answer.]
(c)
The professor that gives the higher proportion of F’s is Professor
[Show the appropriate calculations to support your answer.]
(d)
My score has to be a(n)
% to earn an A for Professor DeGroot.
[Show the appropriate calculations to support your answer.]
.
Activity 15-8:
IQ Scores
mean µ =
and standard deviation ó = __________
(page 339)
(a)
[Sketch the curve.]
(b)
[Shade the area under the curve.]
(c)
The proportion of students with an IQ below 100 is
(d)
The proportion of students with an IQ greater than 130 is
(e)
The proportion of students with an IQ between 110 and 130 is
% .
(f)
With his IQ of 75, Forrest Gump would have a higher IQ than
undergraduates.
% of these
(g)
A person must have an IQ of
[Guess]
Pr ( X < 100) =
% .
% .
in order to be in the top 1% of all IQ’s.
Activity 15-14:
Empirical Rule
(a)
Pr (-1 < Z < 1) =
=
%
(b)
Pr (-2 < Z < 2) =
=
%
(c)
Pr (-3 < Z < 3) =
=
%
(page 341)
According to the “empirical rule,” in any normal distribution, roughly :
(d)
% of the observations fall within
standard deviation of the mean,
% of the observations fall within
standard deviations of the mean,
% of the observations fall within
standard deviations of the mean.
[Draw the curve and the shade the corresponding area under the curve.]
The IQR of the standard normal distribution = 0.67
(Q3)
- (- 0.67) =
.
(Q1)
Therefore the IQR of ANY normal distribution is
* sigma.
And if sigma = 1 (ó = 1), then 1.5 * IQR = 1.5 * (_________) = 2.01
(e)
Outliers are those values lying above
or below
(0.67 + 2.01)
(0.67 - 2.01)
Table II reveals each of these probabilities to be
so the probability of an outlier = 2 *
.
,
=
% .
Activity 15-15:
HINT:
Critical Values
(page 342)
Draw the curve and the shaded area under the curve for each problem.
(a)
If Pr (Z > z*) = .10 , then z* = __________
(b)
If Pr (Z > z*) = .05 , then z* = __________
(c)
If Pr (Z > z*) = .025 , then z* = __________
(d)
If Pr (Z > z*) = .01 , then z* = __________
(e)
If Pr (Z > z*) = .005 , then z* = __________
These values are called the
of the normal distribution.