2.2 Supplemental Handout
NORMAL DISTRIBUTIONS
THE 68-95-99.7 RULE
EXAMPLE 1:
The mean batting average for Major League
Baseball players in 2009 was 0.261 with a
standard deviation of 0.034. Suppose that the
distribution is approximately Normal with
µ = 0.261 and σ = 0.034. N(0.261, 0.034)
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 below 0.227? Show your work
d. What percent of the batting averages are between 0.227 and 0.295? Show your work
e. What percent of the batting averages are between 0.193 and 0.295? Show your work
2.2 Supplemental Handout
NORMAL DISTRIBUTIONS
STANDARD NORMAL DISTRIBUTIONS
ALWAYS SKETCH A CURVE AND SHADE THE APPROPRIATE AREA!
EXAMPLE 2. Using Table A, find the following proportions. Sketch and shade in the appropriate area for
the normal distribution.
a. What is the proportion of observations in
a standard Normal distribution that are
less than 0.81 standard deviation above
the mean?
P (Z < 0.81)
b. What is the proportion of observations in
a standard Normal distribution that are
more than 1.53 standard deviations
above the mean? P (Z > 1.53)
c. What is P (Z < -2.31)?
d. What is P (Z > -1.92)?
e. What is the proportion of observations in
a standard Normal distribution that are
between -1.25 and 0.81?
P (-1.25 < Z < 0.81)
f.
What is P (-0.58 < Z < 1.79)?
2.2 Supplemental Handout
NORMAL DISTRIBUTIONS
Example 3
Use Table A to find the value of z from the standard Normal distribution that satisfies each of the
following conditions. In each case, sketch a standard Normal curve with your value of z(s) marked on the
axis.
a. The 20th percentile
b. 45% of all observations are greater than z
c. The 1st quartile
d. The interquartile range
Using EXAMPLE 2, VERIFY YOUR NUMBERS ARE ACCURATE USING TECHNOLOGY.
Use normalCDF to find the area (proportion)
When getting a proportion below a z-score, use
under the curve
-1E99 (10-99, which is about as close to negative
infinity as your calculator will allow).
When getting a proportion above a z-score, use
1E99 (1099, which is about as close to positive
infinity as your calculator will allow).
When getting a proportion between two z-scores,
use both z-scores as the lower and upper bounds.
P (Z < 0.81) =
P (Z > -1.92) =
P (Z > 1.53) =
P (-1.25 < Z < 0.81) =
P (Z < -2.31) =
P (-0.58 < Z < 1.79) =
2.2 Supplemental Handout
NORMAL DISTRIBUTIONS
Using EXAMPLE 3, VERIFY YOUR NUMBERS ARE ACCURATE USING TECHNOLOGY.
Use inverse Norm (invNORM) to find a z-score at
a percentile.
Caution:
Unlike normalCDF, invNORM only calculates zscores for area to the LEFT of the percentage.
Therefore, if you are needing to know the area to
the right, subtract that percentage from 1 before
you use invNORM!
The 20th percentile
The 1st quartile
45% of all observations are greater than z
The interquartile range
EXAMPLE 4 – IN CONTEXT OF A PROBLEM
In the 2008 Wimbledon tennis tournament, Rafael Nadal averaged 115 miles per hour (mph) on his first
serves. Assume that the distribution of his fist-serve speeds in Normal with a mean of 115 mph and a
standard deviation of 6 mph.
i.
About what proportion of his first serves would you expect to exceed 120 mph?
ii.
What percent of Rafael Nadal’s first serves are between 100 and 110 mph?
iii.
What is the interquartile range of the speed of Rafael Nadal’s first serves?
2.2 Supplemental Handout
NORMAL DISTRIBUTIONS
FREE RESPONSE – 2011 #1
A professional sports team evaluates potential players for a certain position based on two main
characteristics: speed and strength.
(a) Speed is measured by the time required to run a distance of 40 yards, with smaller times indicating
more desirable (faster) speeds. From previous speed data for all players in this position, the times to run
40 yards have a mean of 4.60 seconds and a standard deviation of 0.15 seconds, with a minimum time of
4.40 seconds, as shown in the table below.
Time to run 40 yards
Mean
4.60 seconds
Standard Deviation
0.15 seconds
Minimum
4.40 seconds
Based on the relationship between the mean, standard deviation, and minimum time, is it reasonable to
believe that the distribution of 40-yard running times is approximately normal? Explain.
(b) Strength is measured by the amount of weight lifted, with more weight indicating more desirable
(greater) strength. From previous strength data from all players in this position, the amount of weight
lifted has a mean of 310 pounds and a standard deviation of 25 pounds, as shown in the table below.
Amount of weight lifted
Mean
310 pounds
Standard Deviation
25 pounds
Calculate and interpret the z-score for a player in this position who can lift a weight of 370 pounds.
(c) The characteristics of speed and strength are considered to be of equal importance to the team in
selecting a player for the position. Based on the information about the means and standard deviations of
the speed and strength data for all players and the measurements listed in the table below for Players A
and B, which player should the team select if the team can only select one of the two players? Justify your
answer.
Time to run 40 yards
Amount of weight lifted
.
Player A
4.42 seconds
370 pounds
Player B
4.57 seconds
375 pounds