Transcript Ch 2 The Normal Distribution

Ch 2
The Normal Distribution
2.1 Density Curves and the
Normal Distribution
2.2 Standard Normal Calculations
• We have a clear strategy for exploring data
on a single quantitative variable
– Plot Data, usually Histogram or Stemplot
– Calculate numerical summaries
– Describe the CUSS
• New step:
– If the overall pattern is very regular (not
necessarily symmetric), we can describe it with
a smooth curve
Density Curves
• Easier to work with a smooth curve than with
a histogram
• The curve describes what proportions of the
observations fall within each range of values
• Total area under the curve is exactly 1
• Always on or above the horizontal axis
Density Curves
The shaded area is the proportion of
observations taking values between 7 and 8
Median and Mean of
Density Curve
• Median
– Equal-Areas Point. Half the area to its left and half its area to its
right
– Quartiles divide area into quarters
• Mean
– Balance point at which the curve would balance if it were made
of solid material
– Pulled towards skewing
At which of these points on each curve do the mean
and median fall?
Median: B
Median: A
Median: B
Mean: C
Mean: A
Mean: A
New Notation
• Density Curve is an idealized description of data.
Must distinguish between mean and standard
deviation of a density curve versus those of
actual observations
• Actual Observations:
• Idealized distributions:
mean: 
standard deviation: 
Normal Distributions
• Normal Distributions:
– Symmetric, Single-Peaked, Bell-Shaped
– Always described by giving
mean: 
standard deviation:  (occur at inflection points)
“Empirical” or “68-95-99.7” Rule
For any normal distribution:
• 68% of the observations fall within 1
standard deviation of the mean
• 95% of the observations fall within 2
standard deviations of the mean
• 99.7% of the observations fall within 3
standard deviations of the mean
• 2.6 MEN’S HEIGHTS The
distribution of heights of adult
American men is approximately
normal with mean 69 inches and
standard deviation 2.5 inches.
Draw a normal curve on which this
mean and standard deviation are
correctly located.
2.2 Standard Normal Calculations
N ( , )
• We can “standardize” all normal distributions by
measuring in units of size 
about the
mean 
• Standardizing Observations

If x is an observation from a distribution that has mean
and standard deviation
, then standardized value of x is

z
x

Called z-score. The z-score tells us how many standard
deviations the original observation falls from the mean and
in which direction
Standard Normal Distribution
N(1,0)
Standard Normal Table (Table A)
• Ex: Find the proportion of observations
from the standard normal table that are
less than 2.22
0.9868
• Ex: The heights of young women are
approximately normal N(64.5”,2.5”). Find the
standardized height of a woman who is 68” tall.
z
x

68  64.5
z
2.5
z  1.4
• What proportion of women are less than
68” tall?
About 0.9192 or 91.92% of
women are less than 68”
tall.
• Ex: Cholesterol Level for 14 year old boys
N(170,30). Levels above 240 may require
medical attention. Units: cholesterol per deciliter
of blood.
– What % of 14 yr. old boys have more than 240 mg/dl
cholesterol?
240  170
z
30
z  2.33
• Looking for proportions of values above 2.33.
• Table A gives proportions below a z score. Can we still use
the table to answer the question?
Proportion of 14 yr. old boys with a cholesterol level greater than 240 is
approx. .0099 or .99%
• What proportion of 14 yr old boys have blood
cholesterol between 170 and 240 mg/dl?
• Looking for: 170  x  240
• Standardize both scores:
0  z  2.33
The proportion of boys have
blood cholesterol between 170
and 240 is 0.4901 or about
49%
• EX: Scores on the SAT Verbal follow N(505,110).
To earn in the top 10% how high must a student
score?
Closest to 0.9 is 0.8997
which corresponds to
z=1.28
x  505
 1.28
110
x = 646….does this make
sense in the context of
the problem?
Assessing Normality
• Construct a histogram or stem-plot to verify
bell shape
• Calculate the percent of observations within
1 and 2 standard deviations from the mean
and compare to empirical rule
•
Construct a normal probability plot using
your calculator. If data is close to normal,
plotted points will lie in close to a straight
line.
2.26 CAVENDISH AND THE DENSITY OF THE EARTH Repeated careful
measurements of the same physical quantity often have a distribution that
is close to normal. Here are Henry Cavendish’s 29 measurements of the
density of the earth, made in 1798. (The data give the density of the earth
as a multiple of the density of water.)
5.50 5.61 4.88 5.07 5.26 5.55 5.36 5.29 5.58 5.65
5.57 5.53 5.62 5.29 5.44 5.34 5.79 5.10 5.27 5.39
5.42 5.47 5.63 5.34 5.46 5.30 5.75 5.68 5.85
(a) Construct a stemplot to show that the data are reasonably symmetric.
(b) Now check how closely they follow the 68–95–99.7 rule. Find and s,
then count the number of observations that fall between – s and s,
between – 2s and 2s, and between – 3s and 3s. Compare the percents of
the 29 observations in each of these intervals with the 68–95–99.7 rule.
(c) Use your calculator to construct a normal probability plot