Download "Statistical thinking will one day be as necessary for efficient

"Statistical thinking will one day be as necessary for efficient citizenship as the ability to
read and write." -- H.G. Wells, 1866-1946
2.1 DENSITY CURVES AND NORMALDISTRIBUTIONS (Pages 64-79)
OVERVIEW: Sometimes the overall pattern of a distribution is such that we can describe
it with a smooth curve. It is remarkable how many natural phenomena appear to be
related to a bell-shaped curve known as a normal distribution. When appropriate, using a
normal distribution model to represent distributions that occur in real-life situations can
be extremely useful in statistical analysis.
Density curve:
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A histogram becomes a density curve if the scale is adjusted so that the total area of the bars is 1
sq. unit.
The __________ of a density curve is the point that divides the area under the curve into halves.
The __________ of a density curve is the "balance point" of the curve. (Think of a teeter-totter.)
As an illustration, consider the set {1,2,3,5,11,14}. If each box with an X has an area of
1/6, then the total area of the six boxes would be 1. The median of this set is ____, and
the mean (the balance point of the teeter-totter) is _____.
XXX
X
X
X
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A special type of density curves form ___________________. These distributions are symmetric, single
peaked, and bell-shaped. A normal curve is determined by the mean () and standard deviation () of
the data set. This is notated by N(  ,  ) .
In the normal distribution we have a common property we use called the ______________ which says,
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A normal distribution curve has two points where curvature changes. These are called
____________________, and they are located 1 standard deviation on either side of the mean.
On the TI-83, 2nd/Vars/normalcdf(lowerbound,upperbound,mean,standarddeviation) can be very useful
in statistical analysis. Note that if a normal distribution has mean = 0 and st.dev. = 1, then
normalcdf(________) = .6826894809
normalcdf(________) = .954499876
normalcdf(________) = .9973000656
An observation's __________ is the percent of the distribution that is at or to the left of the observation.
If, for instance, if you have a test score representing the 90th percentile, then only 10% of the test-takers
scored higher than you did.
The distribution of heights for women 18-24 is approximately normal with   64.5,   2.5.
1. What % of women are shorter than 62”?
2. Between what heights do the middle 68% fall?
3. What percentile does a height of 69.5” correspond to?
4. The mean grade on the 1998 AP Statistics exam of school A was 3.45 with a std. deviation of .3. The
mean grade on the same test at school B was 3.45 with a std. deviation of .8. Assuming that the
numbers of students are approximately the same and that both distributions are symmetric and moundshaped, describe any differences you would expect to see in the graphs of the distributions.
5. A set of 2,000 measurements has a symmetric, mound-shaped distribution. The mean is 5.3 and the
standard deviation is 0.7. Determine an interval that contains approximately 1,360 data values.
Extremely important for success of the Advanced Placement Statistics Examination:
If you are given a numerical data set, always (I repeat, always) display the shape of the
distribution.
Using the TI-83, this can be done very easily with a histogram or a box plot.