Download Section 2.2 Density Curves and Normal Distributions

Section 2.2
Density Curves and Normal Distributions
Batting averages
The histogram below shows the distribution of batting average (proportion of
hits) for the 432 Major League Baseball players with at least 100 plate
appearances in a recent season. The smooth curve shows the overall shape of the
distribution.
In the first graph below, the bars in red represent the proportion of players who
had batting averages of at least 0.270. There are 177 such players out of a total
of 432, for a proportion of 0.410. In the second graph below, the area under the
curve to the right of 0.270 is shaded. This area is 0.391, only 0.019 away from
the actual proportion of 0.410.
In general, we will use Greek letters to represent the "true"
value for a population and non-Greek letters to represent an
"estimated" value for a sample
Mean
Sample
Population
μ
Standard
Deviation
Sx
σ
In the previous example about batting averages for Major League Baseball
players, the mean of the 432 batting averages was 0.261 with a standard
deviation of 0.034. Suppose that the distribution is exactly Normal with =
0.261 and = 0.034.
Problem:
(a) Sketch a Normal density curve for this distribution of batting averages.
Label the points that are 1, 2, and 3 standard deviations from the mean.
(b) What percent of the batting averages are above 0.329? Show your work.
(c) What percent of the batting averages are between 0.193 and 0.295? Show
your work.
Batting averages
How well does the 68–95–99.7 rule apply to the distribution of batting averages
we encountered earlier? About 67.6% of the batting averages were within one
standard deviation of the mean, slightly less than the 68% we would expect from
a Normal distribution. About 96.3% of the batting averages were within two
standard deviations of the mean, slightly more than the 95% we would expect
from a Normal distribution. Finally, about 99.8% of the batting averages were
within 3 standard deviations of the mean—quite close to the 99.7% we expected.
Working Backward
In a standard Normal distribution, 20% of the observations are above what
value?
Serving speed
In a recent tournament, tennis player Rafael Nadal averaged 115 miles per hour
(mph) on his serves. Assume that the distribution of his serve speeds is Normal
with a standard deviation of 6 mph.
Problem: About what percent of Nadal’s serves would you expect to exceed
120 mph?
Serving speed (continued)
Problem: What percent of Rafael Nadal’s serves are between 100 and 110
mph?
Heights of three-year-old females
According to http://www.cdc.gov/growthcharts/, the heights of three-year-old
females are approximately Normally distributed with a mean of 94.5 cm and a
standard deviation of 4 cm.
Problem: What is the third quartile of this distribution?
No space in the fridge?
The measurements listed below describe the usable capacity (in cubic feet) of a
sample of 36 side-by-side refrigerators (Consumer Reports, May 2010).Are the
data close to Normal?
12.9 13.7 14.1 14.2 14.5 14.5 14.6 14.7 15.1 15.2 15.3 15.3
15.3 15.3 15.5 15.6 15.6 15.8 16.0 16.0 16.2 16.2 16.3 16.4
16.5 16.6 16.6 16.6 16.8 17.0 17.0 17.2 17.4 17.4 17.9 18.4
Problem: Use the histogram and Normal probability plot below to determine
if the distribution of areas for the 50 states is approximately Normal.
pg. 128
33,35,39,42,44,45,48,50,51
pg. 130
53,55,57,59
pg.130
54,63,65,66,67,69-74