Download Unit 2 - AP Statistics - Lang

Unit 2: Modeling Distributions
of Data
Homework Assignment
• For the A: 1, 3, 5, 9 - 23 Odd, 25 – 30, 33,
35, 39 – 59 Odd and 54, 63, 65 – 67, 69 –
74, R1 – R11
• For the C: 1, 3, 5, 9, 11, 15, 19, 21, 23, 33,
35, 41, 45, 49, 53, 54, 57, 59, 63, 65, 67,
R1 – R 11
• For the D- : 1, 9, 11, 19, 21, 33, 35, 45, 49,
54, 57, 63, 67, R1 – R 11
Density Curves
• Histograms cannot properly show the
distribution of a continuous variable
• Density curves can!
• Area = probability!!
…continued
• Properties
– Total area under the curve is 1
– Curve is above the x-axis
• Note: since a vertical line has no area, a
density curve cannot tell us the probability
of obtaining a single value
Some Examples
Density Curves Recap…
•
•
•
•
Can be created by smoothing histograms
ALWAYS on or above the horizontal axis
Has an area of exactly one underneath it
Describes the proportion of observations that fall
within a range of values
• Is often a description of the overall distribution
• Uses m & s to represent the mean & standard
deviation
Example 01
• Is this a valid density curve?
Example 02
• Is this a valid density curve?
Example 03
• Is this a valid density curve?
Example 04
• Is this a valid density curve?
The Normal Distribution
• Most important distribution of all
• Symmetric
• Two Parameters:
– Mean (center)
– Standard Deviation
• There is a trick for finding the standard
deviation!
Self Check #8
Assignment #4
The Normal Distribution
• Most important distribution of all
• Symmetric
• Two Parameters:
– Mean (center)
– Standard Deviation
• There is a trick for finding the standard
deviation!
The Points of Inflection
• As you trace your finger along the graph,
something changes at the indicated points
…continued
• These are called Points of Inflection
• They are the spots where a graph changes
from rising/falling steeply to rising/falling
gently
• The distance from the center to the xcoordinate of a POI is the standard
deviation
• More on this in Calculus!
Empirical Rule
• Approximately 68% of the
observations are within 1s of m
• Approximately 95% of the
CanofONLY
observations are within 2s
m be used
with normal curves!
• Approximately 99.7% of the
observations are within 3s of m
Normal Curve recap…
• Bell-shaped, symmetrical curve
• Transition points between cupping
upward & downward occur at m + s
and m – s
• As the standard deviation increases,
the curve flattens & spreads
• As the standard deviation decreases,
the curve gets taller & thinner
Self Check #9
Multiple Choice Test #3
The Standard Normal
• μ=0; σ=1
• Any normal distribution can be transformed
into a Standard Normal by standardizing
z 
x  mx
sx
Using the Z-Table
• Units and
tenths down
the side
• Hundredths
along the top
• Left area at
the
intersection
Questions
• What if you want area to the right?
• What if you want the area between two
values of z?
• What if your z doesn’t appear on the chart?
• What if you are given the area, but want to
find the value of the variable?
• Do you know how to use the calculator to
bypass the table entirely?
What do these z scores mean?
-2.3
1.8
6.1
-4.3
2.3 s below the mean
1.8 s above the mean
6.1 s above the mean
4.3 s below the mean
Jonathan wants to work at Utopia
Landfill. He must take a test to see if
he is qualified for the job. The test
has a normal distribution with m = 45
and s = 3.6. In order to qualify for the
job, a person can not score lower than
2.5 standard deviations below the
mean. Jonathan scores 35 on this test.
Does he get the job?
No, he scored 2.78 SD below the mean
Sally is taking two different math
achievement tests with different means
and standard deviations. The mean score
on test A was 56 with a standard deviation
of 3.5, while the mean score on test B was
65 with a standard deviation of 2.8. Sally
scored a 62 on test A and a 69 on test B.
On which test did Sally score the best?
She did better on test A.
Example 05
• Find the probability that the variable takes a
value greater than 1.
Example 06
• Find the probability that the variable takes a
value between
1.4 and 2.
Self Check #10
z score recap…
• Standardized score
• Creates the standard normal density curve
• Has m = 0 & s = 1
z 
x  mx
sx
Calculator Activity
The height of male students at BHS
is approximately normally
distributed with a mean of 71 inches
and standard deviation of 2.5 inches.
a) What percent of the male students
are shorter than 66 inches? About 2.5%
b) Taller than 73.5 inches? About 16%
c) Between 66 & 73.5 inches?About 81.5%
Example 07
• In 2001, the reading scores on the ACT
were approximately normally distributed
with mean 21.3 and standard deviation 6.
• What proportion of reading scores were less
than 9.3?
• What proportion of reading scores were
between 15.3 and 27.3?
Example 08
• Final grades in a college class are
(historically) normally distributed with
mean 72 and standard deviation 12.5.
• What proportion of final grades are at least
97?
• What proportion of final grades are between
34.5 and 59.5?
Example 09
• In 2002, salaries for human resource clerks
were approximately normally distributed
with mean $27,937 and standard deviation
$1700.
• What percentage of salaries were greater
than $30,000?
• What percentage of salaries were between
$20,000 and $25,000?
Self Check #11
Finding a value given a
proportion
• We may instead want to find the observed value
with a given proportion of the observations above
or below it.
• To do this, use Standard Normal (Z) Table
backward.
• Find the given proportion in the body of the table,
read the corresponding z from the left column and
top row, then “un-standardize” to get the observed
value.
Example 10
• The amount of soda in particular size bottle varies
normally with mean 32 oz and standard deviation
0.55 oz.
• What percentage of bottles contain less than 31 oz?
• The lowest 5% of volumes are
less than _____.
• The greatest 1% of volumes are
more than ____.
Calculator Activity
Interpreting Center Activity
Self Check #12
Assignment #5
Assessing Normality
• We need to develop methods for checking if
the population is normal.
• One method we already know:
a. create a histogram or stemplots.
b. Look for non-normal features such as
outliers, skewness, gaps, or clusters
Normal Probability Plot
• A normal probability plot provides a good
assessment of the adequacy of the normal
model for a set of data.
• Any normal distribution produces a straight
line on the plot because standardizing is a
transformation that can change the slope
and intercept of the line in our plot but
cannot change a line into a curved pattern.
Data Sampled From a Normal
Distribution
Notice that the normal probability plot (NPP) is
basically straight. That's the idea: Normal data =
straight NPP. So, when the NPP is straight you have
evidence that the data is sampled from a normal
distribution.
Data Sampled From a Right
Skewed Distribution
For right skewed data, the normal probability plot is
generally not straight. In general this sort of curvature
in the NPP implies right skew.
Data Sampled From a Left
Skewed Distribution
For left skewed data, the normal probability
plot is generally not straight. In general this
sort of curvature in the NPP implies left skew.
Normal Probability Plot Recap
• As you progress in the course normal
probability plots will become clearer. At
this point you should be able to recognize
whether the sample data appears to be from
a population that is normal based on the
normal probability plot.
Self Check #13
Multiple Choice Test #4