Download 1.3 Density curves p50 •Some times the overall pattern of a large

1.3 Density curves p50
•Some times the overall pattern of a large
number of observations is so regular that we
can describe it by a smooth curve.
• It is easier to work with a smooth curve,
because the histogram depends on the choice
of classes.
•Density Curve (p52)
Density curve is a curve that
- is always on or above the
horizontal axis.
- has area exactly 1 underneath it.
•A density curve describes the overall
pattern of a distribution.
1
•Example: The curve below shows the
density curve for scores in an exam and
the area of the shaded region is the
proportion of students who scored
between 60 and 80.
2
•The median of a distribution is the
point that divides the area under the
curve in half. (p54)
•A mode of a distribution described by a
density curve is a peak point of the
curve. (p53)
•Quartiles of a distribution can be
roughly located by dividing the area
under the curve into quarters as
accurately as possible by eye.(p53)
3
(Hint: Recall the formula to calculate the area of a triangle)
Normal distributions p54
4
•An important class of density curves are the
symmetric unimodal bell-shaped curves
known as normal curves. They describe
normal distributions.
•All normal distributions have the same
overall shape.
•The density curve for a particular normal
distribution is specified by giving the mean
 and the standard deviation .
•The mean is located at the center of the
symmetric curve and is the same as the
median (and mode).
•Changing  without changing  moves the
normal curve along the horizontal axis
without changing spread.
5
•The standard deviation  controls the
spread of a normal curve. P55
•There are other symmetric bell- shaped
density curves that are not normal. P55
•The normal density curves are specified by
a particular equation. The height of a normal
density curve at any point x is given by
1
 2
2
x


1 
e 2














6
•The 68-95-99.7 rule p56
In the normal distribution with mean
m and std. deviation s,
 Approx. 68% of the observations fall
within  of the mean µ.
Approx. 95% of the observations fall
within 2 of the mean µ.
Approx. 99.7% of the observations fall
within 3 of the mean µ.
•Example (1.34, p57)
The distribution of heights of women
aged 18-24 is approx. normal with mean
µ = 64.5 inches and std. dev.
 = 2.5 inches.
2 = 5 inches. The 68-95-99.7 rule
says that the middle 95% (approx.) of
women are between 64.5-5 to 64.5+5
inches tall.
The other 5% have heights outside
the range from 64.5-5 to 64.5+5 inches .
7
2.5% of the women are taller than
64.5+5 .
Ex. 1) The middle 68% (approx.) of
women are between ____ to ___ inches
tall.
2) ___% of the women are taller than 67
3) ___% of the women are taller than 72
8
•Notation: A normal distribution with
mean  and std. dev.  is denoted by
N(,).
•The distribution of women’s heights is
N(64.5, 2.5).
•Standardizing and z-scores p58
If x is an observation from a
distribution that has mean  and std. dev.
 , the standardized value of x is
  A standardized value is often
z  x
called a z-score.
A z-score tells us how many standard
deviations the original observation falls
away from the mean.
9
•Example (1.35 p58)
heights of women are approx.. normal
with mean  = 64.5 inches and std. dev.
 = 2.5 inches.
The standardized height is
z  height 64.5 The standardized value
2.5
(z-score) of height 68 inches is
z  6864.5 1.4 or 1.4 std. dev. above
2.5
the mean.
A woman 60 inches tall has standardized
height
z  60 64.5 1.8 or 1.8 std. dev.
2.5
below the mean
10
The Standard Normal distribution p59
•The standard normal distribution is
the normal distribution N(0, 1).
•If a random variable X has normal
distribution N(, ), then the
  has the
standardized variable z  x
standard normal dist.
11
The standard normal tables p61
Examples
What proportion of observations of a
std. normal variable Z takes values
a) less than 1.4?
b) greater than 1.4?
c) greater than 1.96
d) between 0.43 and 2.15
e) between –0.92 and 1.43
12
Ex. X~N(65, 15). What proportion of
observations of X takes values?
•less then 50
•greater than 80
•between 50 and 80
13
•Example
Scores on SAT verbal test follow
approx. the N(505, 110) dist. How high
must a student score in order to place in
the top 10% of all students taking the
SAT?
14
Normal quantile plots p65
•A histogram or stem plot can reveal
distinctly nonnormal features of a
distribution.
•If the stemplot or histogram appears
roughly symmetric and unimodal, we use
another graph, the normal quantile plot
as a better way of judging the adequacy
of a normal model
•Use of normal quantile plots. P65
If the points on a normal lie close to a
straight line, the plot indicates that the
data are normal.
Outliers appear as points that are far
away from the overall pattern of the plot.
15
Histogram and the nscores plot for data
generated from a normal distribution (
N(500, 20)) (for 1000 observations)
16
17
Histogram and the nscores plot for data
generated from a right skewed distribution
18
19
Histogram and the nscores plot for data
generated from a left skewed distribution
20
21