Download 2.2 Modeling Distribution of Data

2.2 Modeling Distributions of Data
Date: __________
Normal Distributions
One particularly important class of density curves are the Normal curves, which describe Normal distributions.
All Normal curves are:
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A Specific Normal curve is described by:
Here are two examples of normal curves showing the mean  and standard deviation  .
Definition:
A Normal distribution is described by a Normal density curve. Any particular Normal distribution is
completely specified by:
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We abbreviate the Normal Distribution with mean  and standard deviation  : __________________.
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Properties of all Normal Curves:
“The Empirical Rule” also known as the _____________________________________________.
 Approximately __________ % of the data is within _______ of  .
 Approximately __________ % of the data is within _______ of  .
 Approximately __________ % of the data is within _______ of  .
Example: The batting averages for Major League Baseball players in 2009 is 0.261 with a standard deviation of 1.034.
Suppose the distribution is exactly Normal.
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.
The Standard Normal Distribution
All Normal distributions are the same if we measure in units of size σ from the mean µ as center.
Definition:
All Normal distributions are the same if we measure in units of size σ from the mean µ as center.
The _________________________ _________________ _____________________ is the Normal distribution with
mean 0 and standard deviation 1.
If a variable x has any Normal distribution N(µ,σ) with mean µ and standard deviation σ, then the standardized
variable
z
has the standard Normal distribution _________________.
Because all Normal distributions are the same when we standardize, we can find areas under any Normal curve from
a single table. In our book, this is found in table A. Table A is a table of areas under the standard Normal curve. The
table entry for each value z is the area under the curve to the ________ of z.
Example: Suppose we want to find the proportion of observations from the standard Normal distribution that are less
than 0.81. We can use Table A:
1)
P( z  0.81)  __________
z
.00
.01
.02
0.7
.7580
.7611
.7642
0.8
.7881
.7910
.7939
0.9
.8159
.8186
.8212
2) Use Table A to determine the proportion of observations from the standard Normal distribution that are between 1.36 and 0.74.
How to Solve Problems Involving Normal Distributions:
 State:
 Plan:
 Do:
 Conclude:
3) When Tiger Woods hits his driver, the distance the ball travels can be described by N(304, 8). What percent of
Tiger’s drives travel between 305 and 325 yards?
4) In the figure below, what is z?
Assessing Normality:
The Normal distributions provide good models for some distributions of real data. Many statistical inference procedures
are based on the assumption that the population is approximately normally distributed. Consequently, we need a
strategy for assessing Normality.
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Your graphing calculator can construct a ________________ _______________________ ___________. This plot is
constructed by plotting each observation in a data set against its corresponding percentile’s z-score.
A3: Read 2.2 and do 41, 43, 45, 47, 49, 51
A4: 53, 55, 57, 59