Download File

Bell Ringer
If the test average for your class is
78 with a standard deviation of 5,
what would your z-score be if you
made an 88?
Boxplot of 3rd Hour, 4th Hour
100
90
Data
80
70
60
50
3rd Hour
4th Hour
Standardizing values means we shift the data
to fit a normal model (bell-shaped curve).
This is only appropriate for unimodal
and symmetric distributions.
Recall: A standardized value is called a z-score.
 Standardizing into z-scores does not change
the shape of the distribution.
 Standardizing into z-scores changes the
center to 0.
(Also called the 68-95-99.7 Rule)
 About 68% of observations fall within 1 standard
deviation of the mean.
 About 95% of observations fall within 2 standard
deviations of the mean.
 About 99.7% of observations fall within 3 standard
deviations of the mean.
This means that almost all z-scores
will be between -3 and 3.
But we need to remember that all data
make up 100%
The Normal Model split into sections:
(The total area under the curve is 100%)
Only 2.5% of data
will have a z-score
greater than 2.
2.5%
𝜎
2.5%
𝜎
𝜎
𝜎
𝜎
𝜎
Similarly 2.5% will
have a z-score less
than -2.
 A number describing a population is
called a parameter.
 A number describing a sample (just a part
of the population) is called a statistic.
Statistic
𝑥 = 𝑚𝑒𝑎𝑛
𝑆 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
Parameter
𝜇 = 𝑚𝑒𝑎𝑛
𝜎 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝑇ℎ𝑒 𝑛𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝑁 𝜇, 𝜎 𝑚𝑒𝑎𝑛𝑠 𝑡ℎ𝑒 𝑁𝑜𝑟𝑚𝑎𝑙 𝑚𝑜𝑑𝑒𝑙 𝑤𝑖𝑡ℎ
𝑎 𝑚𝑒𝑎𝑛 𝑜𝑓 𝜇 𝑎𝑛𝑑 𝑎 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝜎.
This model is N(184,8)
160
168
176
184
192
200
208
𝐶𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡 𝑡ℎ𝑒 𝑁𝑜𝑟𝑚𝑎𝑙 𝑚𝑜𝑑𝑒𝑙 𝑓𝑜𝑟 𝑁(50, 7)
29
36
43
50
57
64
71
Example:
In the 2006 Winter Olympics men’s combined event, Jean-Baptiste
Grange of France skied the slalom in 88.46 seconds – about 1
standard deviation faster than the mean. If a Normal model is useful
in describing slalom times (meaning the data was nearly symmetric),
about how many of the 35 skiers finishing the event would be
expected to have skied the slalom faster than Jean-Baptiste?
His time lies at about 1 standard deviation below the mean.
That means approximately 16% of the times were lower. 16%
of 35 is 5.6, so probably 5 or 6 skiers did better.
As a group, the Dutch are among the tallest people in the world. The
average Dutch man is 184 cm tall – just over 6 feet. If a Normal
model is appropriate and the standard deviation for men is about 8
cm, what percentage of all Dutch men will be over 2 meters (200
cm) tall?
160
168
176
184
192
200
208
About 2.5% of all Dutch
men will be over 2
meters tall.
Practice:
Suppose it takes you 20 minutes, on average, to drive to school
with a standard deviation of 2 minutes. Also suppose the Normal
model is appropriate for the distribution of driving times.
a) How often will you arrive at school in less than 22 minutes?
b) How often will it take you more than 24 minutes?
c) Do you think the distribution of your driving times is unimodal
and symmetric?
d) What does this say about the accuracy of your predictions?
Answers:
a) 84% of the driving times will be less than 22 minutes.
b) 2.5% of the driving times will be more than 24 minutes.
c) Road blocks, stop lights, wrecks, and other incidents
could cause some driving times to be delayed. It is
possible for the distribution to end up skewed to the
right.
d) If it is skewed, the Normal model would not be
appropriate and our predictions would not be accurate.
Today’s Assignment:
Pg. 131 #25, 26