Download Normal Distributions

2.1 Normal Distributions
AP Statistics
When asked to examine a set of univariate,
quantitative data:
– make a graph
– Look at the overall pattern
– Use numbers to describe the center and spread
When the data is “regular” enough, the overall
pattern can be described by a curve, or
mathematical model.
Because the shape of a histogram changes
according to the size of the classes we choose,
replacing it with a smooth curve eliminates
the need for those choices. When the area
under that curve is exactly one, the areas
under the curve represent proportions of the
total area.
Any curve under which the area is exactly one is
called a density curve.
Page 83 #2.2
Figure 2.7 displays the density curve of a uniform distribution. The curve takes the constant value
1 over the interval from 0 to 1 and is zero outside the range of values. This means that data
described by this distribution take values that are uniformly spread between 0 and 1. Use areas
under this density curve to answer the following questions:
a)
Why is the total area under this curve equal to 1?
b)
What percent of the observations lie above 0.8?
c)
What percent of the observations lie below 0.6?
d)
What percent of the observations lie between 0.25 and 0.75?
e)
What is the mean of this distribution?
Page 83 #2.2
Figure 2.7 displays the density curve of a uniform distribution. The curve takes the constant value
1 over the interval from 0 to 1 and is zero outside the range of values. This means that data
described by this distribution take values that are uniformly spread between 0 and 1. Use areas
under this density curve to answer the following questions:
a)
Why is the total area under this curve equal to 1?
The area under the curve is a rectangle with height 1 and width 1. Thus the total area
under the curve = 1 x 1 = 1
b)
What percent of the observations lie above 0.8?
20% (The region is a rectangle with height 1 and base width 0.2; hence the area is 0.2)
c)
What percent of the observations lie below 0.6?
d)
e)
60%
What percent of the observations lie between 0.25 and 0.75?
What is the mean of this distribution?
Mean = ½ of 0.5, the “balance point” of the density curve.
50%
Page 83-84 #2.3
A line segment can be considered a density “curve,” as shown in Exercise 2.2. A
“broken line” graph can also be considered a density curve. Figure 2.8 shows such a
density curve.
(a) Verify that the graph in Figure 2.8
is a valid density curve.
For each of the following, use areas under this density curve to find the
proportions of observations within the given interval:
(b) 0.6 < X < 0.8
(c) 0 < X < 0.4
(d) 0 < X < 0.2
(e) The median of this density curve is a point between X = 0.2 and X = 0.4.
Explain why.
Page 83-84 #2.3
A line segment can be considered a density “curve,” as shown in Exercise 2.2. A
“broken line” graph can also be considered a density curve. Figure 2.8 shows such a
density curve.
(a)
Verify that the graph in Figure 2.8
is a valid density curve.
For each of the following, use areas under this density curve to find the proportions of
observations within the given interval:
(b) 0.6 < X < 0.8
(c)
(d)
(e)
0.2
0 < X < 0.2 0.6
0.35
0 < X < 0.4
The median of this density curve is a point between X = 0.2 and X = 0.4. Explain
why.
The median is the “equal-areas” point. By (d), the area between 0 and 0.2
is 0.35. The area between 0.4 and 0.8 is 0.4. Thus the “equal-areas” point
must lie between 0.2 and 0.4
Page 113 #2.38
A certain density curve consists of a straight-line segment that begins at the
origin, (0, 0), and has slope 1.
(a) Sketch the density curve. What are the coordinates of the right endpoint
of the segment? (Note: The right endpoint should be fixed so that the
total area under the curve is 1. This is required for a valid density curve.)
(b) Determine the median, the first quartile (Q1), and the third quartile (Q3).
(c) Relative to the median, where would you expect the mean of the
distribution?
(d) What percent of the observations lie below 0.5? Above 1.5?
Page 113 #2.38
A certain density curve consists of a straight-line segment that begins at the origin, (0, 0), and has
slope 1.
(a) Sketch the density curve. What are the coordinates of the right endpoint of the
segment? (Note: The right endpoint should be fixed so that the total area under
the curve is 1. This is required for a valid density curve.)
(b) Determine the median, the first quartile (Q1), and the third quartile (Q3).
median = 1.
Q1 = .707
Q3 = 1.225
(c) Relative to the median, where would you expect the mean of the distribution?
The mean will lie to the left of the median because the density curve is
skewed left.
(d) What percent of the observations lie below 0.5? Above 1.5?
12.5% of the observations lie below 0.5.
None (0%) of the observations lie above 1.5.
The most common density curve is the standard
normal distribution model.
Normal distributions are symmetric, bell-shaped
curves. We describe the center and spread by
using the mean and the standard deviation.
Symbolically, we represent a distribution that
is approximately normal using this symbol:
N(µ, σ)
According to the Empirical Rule, in a normal distribution approximately 68%
of the data will lie within one standard deviation of the mean, 95% of the
data will lie within two standard deviations of the mean, and 99.7% of the
data will lie within three standard deviations of the mean.
Calculate the percentages/areas between
each set of bars.