Transcript IB Math SL 24 AB 2014x

NORMAL DISTRIBUTIONS &
PROBABILITIES WITH A
CALCULATOR
24AB
VIDEO
• http://onlinestatbook.com/movies/normal_distributi
on/intro.mp4
• Goodie
BASED ON THE VIDEO,
1. Which are other names for the normal
distribution? Select all that apply.
A. Typical curve
B. Gaussian curve
C. Regular distribution
D. Galileo curve
E. Bell-shaped curve
F. Laplace's distribution
BASED ON THE VIDEO,
2. Select all of the statements that are true
about normal distributions.
A. They are symmetric around their mean.
B. The mean, median, and mode are equal.
C. They are defined by their mean and
skew.
D. The area under the normal curve is equal
to 1.0.
E. They have high density in their tails.
NORMAL DISTRIBUTION
• The normal distribution is the most important and most
widely used distribution in statistics.
• Examples of data that are normally distributed:
• heights & weights of people,
• Scores for tests taken by a large amount of people (ex. SAT)
• dimensions of manufactured goods
CHARACTERISTICS OF A
NORMAL DISTRIBUTION
• Normal distributions are symmetric around their
mean.
• The mean, median, and mode of a normal
distribution are equal.
• The area under the normal curve is equal to 1.0.
• Normal distributions are denser in the center and
less dense in the tails.
• Normal distributions are defined by two parameters,
the mean (μ) and the standard deviation (σ).
• For a normal distribution with mean μ and standard
deviation, σ, the percentage of breakdown of where the
data lies is shown below.
• Notice that:
• ≈ 68.26% of the data lies within 1 standard deviation of the
mean
• ≈ 95.44% of the data lies within 2 standard deviations of
the mean
• ≈ 99.74% of the data lies within 3 standard deviations of
the mean
PROBABILITY DENSITY FUNCTION
• The probability density function of the normal
distribution (the height for a given value on the x
axis) is shown below.
2
1
f ( x) 
e
 2
1  x 
 

2  
where 𝜇 is the mean and 𝜎 is the standard deviation.
Don’t worry if this confuses you, we will not be
referring to it later.
NOTATION
We can use the following notation to describe a set of
data that is normally distributed :
X ~ N ( , )
2
“X is a random variable that is normally distributed with
a mean of 𝝁 and variance of 𝝈𝟐 .”

A random
variable is a
function that
associates a
unique
numerical value
with every
outcome of an
experiment.
EXAMPLE
• The chest measurements of 18 year old male footballers
are normally distributed with a mean of 95 cm and a
standard deviation of 8 cm.
• Represent this information on a normal curve.
• What percent of the data is between 87 and 103 cm?
68.3%
• What percent of the data is greater than 111 cm?
2.28%
EXAMPLE
• The chest measurements of 18 year old male
footballers are normally distributed with a
mean of 95 cm and a standard deviation of 8
cm.
• Find the percentage of footballers with chest
measurements between:
• 87 cm and 103 cm
68.3%
• 103 cm and 111 cm 13.6%
• Find the probability that the chest measurement of a
randomly chosen footballer is between 87 cm and 111
cm.
0.819
• Find the value of k such that approximately 16% of
chest measurements are below k. k = 87 cm
FURTHER EXPLANATION OF LAST
QUESTION
• Find the value of k such that approximately 16% of
chest measurements are below k.
AREA UNDER THE NORMAL CURVE
• The area under a normal curve represents the
proportion of data at (or between) data values.
• The total area under a normal curve is 1. (100%)
x
• Shaded area = proportion of data less than x
= percent of data less than x
NOTATION
The notation below states:
P( X  a)
“The probability that the random
variable is less than or equal to a.”
P ( a  X  b)
“The probability that the random
variable is greater than or equal to a
and less than or equal to b.”

a
A random
variable is a
function that
associates a
unique
numerical value
with every
outcome of an
experiment.
24B: PROBABILITIES WITH A
CALCULATOR
EXAMPLE TIME!
ANOTHER AWESOME EXAMPLE:
INVNORM HELP
• InvNorm calculator help:
https://www.youtube.com/watch?v=2my8dbhjyt4
HOMEWORK – YIPEE!
•24A: Your choice of 5
problems from #1 –
13.
•24B: Your choice of 4
problems from #1 – 8.