Transcript AP-–-5cx

AP - Bellwork
What does Tchebycheff’s Worst-Case Scenario tell us?
ANSWER: P118
Tchebycheff tells us that at least 89% of values lie within
3 standard deviations and that 96% of all values must be
within 5 standard deviations of the mean….whether the
Normal model can be applied or not.
FINDING NORMAL PERCENTILES
HW: READ p119-125 (up to Are you…) DO p132 15, 29,
33, 35, 37
Let’s revisit the car mileage…
Suppose a Normal model describes fuel efficiency of cars
currently registered in your state. The mean is 24 mpg and
standard deviation is 6 mpg.
 Sketch the Normal model and explain what it means.
6
12
18
24
30
36
42
According to the normal model, we expect 68% of cars to get
Btwn 18 and 30 mpg
Btwn 12 and 36 mpg and
________________,
95% of cars to get ______________,
Btwn 6 and 42 mpg
99.7% of cars to get _______________
Car mileage, cont’d…
 Approximate the percent of all cars that get less
than 15 mpg.
6.68%
15
24
15 mpg is 1.5 standard deviations below the
mean…where does the 6.68% come from?
Two methods to finding NORMAL Percentages
1) Z Table (pg A-124)
2) Calculator – normalcdf
-2nd distr
-normalcdf (zLeft , zRight)
(pick a really low, or really high, z value for the left/right)
Ex: for this problem enter: normalcdf (-5 , -1.5)
In context: According to the Normal model, about 6.68% of
cars are expected to get less than 15 mpg.
Car mileage, cont’d…
 What percent of all cars get between 20 and 30 mpg?
58.89 of
According to the Normal model, about ________%
cars are expected to get between 20 and 30 mpg.
Car mileage, cont’d…
 What percent of cars get more than 40 mpg?
0.38
According to the Normal model, about ________%
of
cars are expected to get more than 40 mpg.
Car mileage, cont’d…
 Describe the fuel efficiency of the worst 20% of all cars.
According to the Normal model, the worst 20% of cars have
18.95 mpg.
a fuel efficiency less than about _________
Car mileage, cont’d…
 What gas mileage represents the third quartile?
According to the Normal model, the third quartile of the
28.05 mpg.
distribution of fuel efficiency is about ________
Car mileage, cont’d…
 Describe the gas mileage of the most efficient 5% of all cars.
According to the Normal model, we expect the most efficient 5%
33.87 mpg.
of cars to have fuel efficiency in excess of __________
Car mileage, cont’d…
 What gas mileage would you consider unusual?
Why?
ANSWER: It depends on your definition of unusual. If +/- 2 standard
deviation is considered unusual, then, according to the Normal model, any
gas mileage below 12 mpg or above 36 mpg would be considered unusual.
If +/- 3 standard deviations is considered unusual, the, according to the
Normal model, any gas mileage below 6 mpg or above 42 mpg would be
unusual.
You try….
Consider the fuel economy model again: N(24,6).
Sketch pictures as you answer these FOUR QUESTIONS:
1) What percent of cars get under 20 mpg?
25.25
According to the Normal model, about ___________%
of
cars have gas mileage less than 20 miles per gallon.
You try….
Consider the fuel economy model again: N(24,6).
Sketch pictures as you answer these FOUR QUESTIONS:
2) An ecology group is lobbying for a national goal calling for no more than
10% of all cars to be under 20 mpg. If the standard deviation does not
change what average fuel efficiency must be attained?
According to the Normal model, in order to meet the goal,
27.69 mpg.
the new mean mileage must be at least _________
You try….
Consider the fuel economy model again: N(24,6).
Sketch pictures as you answer these FOUR QUESTIONS:
3) Car manufacturers argue that they cannot raise the average that much –
they believe they can only get to 26 mpg. What standard deviation
would allow them to meet the “only 10% under 20 mpg” goal?
According to the Normal model, in order to meet the goal,
4.68 mpg.
the new standard deviation must be about ________
You try….
Consider the fuel economy model again: N(24,6).
Sketch pictures as you answer these FOUR QUESTIONS:
4)
What change in the fuel economy of cars would achieving that
standard deviation bring about? What are the advantages and
disadvantages?
ANSWER: A lower standard deviation would mean more
consistent gas mileage. This would be
advantageous since the new goal of ‘only 10%
below 20 mpg” would be met. However, it
also means that only 10% of cars are expected
to have gas mileage higher than 32 mpg.
FINDING NORMAL PERCENTILES…
This section will be very important going forward.
Be sure you are reading through and understanding the
examples in your text.
You need to be able to calculate z-scores, standard
deviations, means, and percentiles backward and
forward using the z-score formula and normalcdf.
See me if you are having any difficulties with this topic.