Transcript Empirical Rule - Lindbergh School District

AP Statistics
Monday, August 29, 2011
• Density Curves PowerPoint
• HW: 2.9 – 2.13
• QUIZ THURSDAY!
AP STATS - Chapter 2
Density Curves
and
Normal Probability
Distributions
Sampling Distribution (Histogram) and
Density Curve (Red Curve)
Density Curves




A density curve describes the overall pattern of a
distribution.
We will use the smooth curve to describe what
proportions of the observations fall in each range
of values, not the counts of observations.
Areas under the curve represent proportions of
the observations.
Total area under the curve is exactly 1.
Various Density Curves
Uniform Population Model
Total area under the curve (model)
will always equal 1.
AP Statistics
Tuesday, August 30, 2011
• 68 – 95 – 99.7% Rule
• Example Worksheet
• HW: Worksheet
• QUIZ THURSDAY
What does a population that is
normally distributed look like?
 = 80 and  = 10
X
50
60
70
80
90
100
110
Empirical Rule 68-95-99.7% RULE
68%
95%
99.7%
Empirical Rule — restated
68% of the data values fall within 1 standard
deviation of the mean in either direction
95% of the data values fall within 2 standard
deviation of the mean in either direction
99.7% of the data values fall within 3 standard
deviation of the mean in either direction
Remember values in a data set must appear to
be a normal bell-shaped histogram, dotplot, or
stemplot to use the Empirical Rule!
Empirical Rule
34% 34%
68%
47.5%
47.5%
95%
49.85%
49.85%
99.7%
Average American adult male
height is 69 inches (5’ 9”) tall with a
standard deviation of 2.5 inches.
Empirical Rule-- Let H~N(69, 2.5)
What is the likelihood that a randomly selected
adult male would have a height less than 69 inches?
Answer: P(h < 69) = .50
P represents Probability
h represents one adult
male height
AP Statistics
Wednesday, August 31, 2011
• Review Worksheet
• QUIZ TOMORROW
• OPEN HOUSE TONIGHT
AP Statistics
Thursday, September 1, 2011
• QUIZ
AP Statistics
Friday, September 2, 2011
• Standard Normal Distribution (non calculator)
– Number of finger taps in 1 minute
– Calculate class mean and standard deviation
– Read article about Pujols
• HW: 2.1, 2.3, 2.29, 2.31 – 2.33
• 3 DAY WEEKEND!!
What is a z-score?
Standardized values based on the population
mean (µ) and standard deviation (α).
Z ~ N (0, 1)
# OF STANDARD DEVIATIONS ABOVE (+)
OR BELOW (-) THE MEAN
z
x

Assuming X ~ N(66, 2), use the formula to
calculate the corresponding z-scores for the
x-values of 60, 62, 64, 66, 68, 70, and 72.
z
x

If we don’t know the values of x, but we know
X ~ N(40, 4), then we can calculate the missing
corresponding x-values when the z-score is –3, -2,
-1, 0, 1, 2, and 3. Remember Z ~N(0, 1).
z
x

Example 1
Suppose the average height of freshmen at
LHS is 60 inches with a standard deviation
of 1.5 inches. What is the z-score for a
freshman who has a height of
a. 58 inches?
b. 60.15 inches?
Example 2
Suppose the average height of sophomores at
LHS is 62 inches with a standard deviation of
2 inches. What is the height of the sophomore
(x-value) that corresponds to a
a)z-score = 0?
b)z-score = -2.44?
c)z-score = 3.1?
Example 3
Suppose the average height of juniors at LHS
is unknown but the standard deviation is 2.5
inches. What is the population mean height of
juniors if
a)a junior 66 inches tall
corresponds to a z-score
of -.75?
Example 4
Suppose the height of seniors at LHS is 67
inches but the standard deviation is unknown.
What is the standard deviation knowing
a)a senior 68.5 inches tall
has a corresponding z-score
of .87?
b)a senior 63 inches tall has
a corresponding z-score
of –2.43?
(This resulting population standard deviation should be
different from the answer to a.)
Example 5
An incoming freshman took her college’s
placement exams in French and math. In
French, she scored 82 and in math, 86. The
overall results on the French exam had a
mean of 72 and a standard deviation of 8,
while the mean math score was 76 with a
standard deviation 12. On which exam did
she do better RELATIVE to her classmates?
Using the z table (pink sheet)
Find the following probabilities:
1. P(z < 1)
2. P(z < 0)
3. P(z < 1.5)
4. P(z > 1.5)
5. P(2.3 < z < 3.1)
Why do we use z-scores?
To answer questions such as…
1. Suppose teachers at LHS have an age
distribution X ~ N(40, 8). What is the likelihood
that a randomly selected teacher from this
population would have an age of 24 or
younger?
Why do we use z-scores?
2. Assuming the same distribution exists for
age of teachers at LHS, how likely is it for a
randomly selected teacher from LHS to be
older than 50 years of age?
Why do we use z-scores?
3. Again using the same teacher age
distribution at LHS, what is the probability that
a randomly selected teacher’s age would fall
somewhere between 34 and 50 years of age?
Why do we use z-scores?
4. Again using the same teacher age
distribution at LHS, at what age do 25% of the
teachers fall below?
AP Statistics
Tuesday, September 5, 2011
• Standard Normal Distribution (calculator)
• Turn on / Link “Catalog Help”
2nd VARS (DIST)
2:normalcdf(
3:invNorm(
• HW: Worksheet
(gives you a list of distributions we will use)
lower bound, upper bound [, µ, α]
area [, µ, α]
Area to the LEFT
Z-Scores on the CALCULATOR
1. Suppose teachers at LHS have an age
distribution X ~ N(40, 8). What is the likelihood
that a randomly selected teacher from this
population would have an age of 24 or
younger?
Z-Scores on the CALCULATOR
2. Assuming the same distribution exists for
age of teachers at LHS, how likely is it for a
randomly selected teacher from LHS to be
older than 50 years of age?
Z-Scores on the CALCULATOR
3. Again using the same teacher age
distribution at LHS, what is the probability that
a randomly selected teacher’s age would fall
somewhere between 34 and 50 years of age?
Z-Scores on the CALCULATOR
4. Again using the same teacher age
distribution at LHS, at what age do 25% of the
teachers fall below?
Z-Scores on the CALCULATOR
5. Again using the same teacher age
distribution at LHS, at what age do 10% of the
teachers fall above?