Download Proportion

Honors 5 Day 2
SILENT DO NOW
ON DESK:
Ch 18 Notes
Agenda:
Ch 18 Review Practice
Start HW
DO NOW:
Z-Score Review
Homework due Monday:
Chapter 18 Packet
Review from
Wednesday
•
n
Consider samples of size ______
We found that we can model statistics about
NORMAL
those samples on a ___________curve
Sample proportions can be modeled on the curve
p
√(pq/n)
with a center of ____
and an SD of _______
Sample means can be modeled on the curve with
σ/√(n)
μ
a center of ____
and an SD of _______
Review from
Wednesday
Because samples can be modeled on a normal
curve (a sampling distribution), we can
describe the distribution of the sample statistic
(the proportion or mean) using the…
68 - ___
95 - _____
99.7 rule
•
___
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If we want to find the probability or percentile
Z-SCORE
of a given statistic, we find the ___________
and look up the probability in the chart
Sampling Distributions
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Sampling Distribution of Proportions
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drawing samples from
a given population
and modeling a
given proportion on a
curve to help understand
probabilities
Sampling Distribution of Means
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drawing samples of size n
from a given population,
taking the average of each
sample, and modeling the
desired average on a curve
Notation
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Sampling Distribution Normal Model
N (μ_x , σ_x )
For sample
proportions
=p
(population
proportion)
For sample
means
For sample
means
=μ
=σ/√n
(population
mean)
(population
SD over sqrt
of n)
For sample
proportions
=√(pq/n)
In order to model
samples normally…
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Samples must be…
1. RANDOM (if not random, won’t represent
population)
2. BIG ENOUGH (if too small, won’t represent
population)
3. SMALL ENOUGH (if too big, won’t have
enough samples to make a curve)
4. INDEPENDENT
Conditions for
Modeling Normally
1.
Samples must be a random sample of the
population
2*. *If dealing with proportions, the sample size has to
be big enough!
np ≥ 10 and nq ≥ 10
3. small enough: the sample size, n, must be no
larger than 10% of the population- this is to
ensure independence
n < 10% (population)
4. Independence: one person in the sample cannot be
influenced by another
Normal Model for
Sampling Distribution
_
Proportions
Averages
Center μ_x
p
(population
proportion)
μ
Standard
Deviation σx_
√(pq/n)
σ/√n
Conditions
Big enough
np>10
nq>10
Small enough
n<10%(population)
Small enough
n<10%(population)
Random
(population
average)
(population
standard deviation
divided by sqrt n)
Random
Describing
Distributions
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1. Suppose that 62% of CPS seniors score a
20 or higher on the ACT with a standard
deviation of 5.1. Consider the sample of my
95 students- describe the distribution.
2. Suppose the mean ACT score for CPS
seniors is a 20 with a standard deviation of
5.1. Consider my sample of 83 studentsdescribe how the scores should be
distributed.
DESCRIBE
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**When describing a normal curve,
we use the 68-95-99.7 rule
68% of samples show the ____ to be
between ___ and ___
• 95% of samples show the ____ to be
between ___ and ___
• 99.7% of samples show the ____ to
be between ___ and ___
•
Sample Distribution
Probability Examples:
mean and proportion
Mean:
• Suppose the mean ACT score for CPS seniors
is a 20 with a standard deviation of 5.1.
What’s the probability that my senior classes
(95 students) got a 21 or higher on avg?
•
Proportion:
• Suppose that 62% of CPS seniors score a 20
or higher on the ACT with a standard
deviation of 5.1. What’s the probability that
50% of my senior classes (95 students) got a
20 or higher?
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Consider This…
Of all cars on the interstate, 80%
speed. What proportion of speeders
might we see among the next 50
cars? (that is, draw a normal curve
and describe the 68-95-99.7
proportions)
• **be sure to check conditions first!
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Consider This…
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According to the US News and World
Report, in the population of 4,361
students enrolled in Penn State
bachelor’s degree programs, 811 have
miliatry experience, so p=811/4361 =
18.6%. If we were to randomly select
75 students to participate in a research
study, what is the probability that more
than 25% have military experience?
Consider this…
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Suppose 60% of all voters in Cook County
intend to vote for Clinton in the upcoming
election. A poll is taken, 100 voters are
selected by SRS. Let p* be the proportion of
sampled voters who intend to vote for
Clinton. Draw the Sampling Distribution
Model. What is the probability of p*<0.5?
(That is, what is the probability less than half
of the sampled voters intend to vote for
Clinton? This may lead to an incorrect
prediction of Clinton losing.)
Another Example
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According to a Gallup Poll in 2006,
44% of American households own a
dog. What is the probability that a
random sample of 60 households will
have a sample proportion greater
than 40%?
Consider Means
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The weight of potato chips in a medium
size bag is stated to be 10 ounces. The
amount that the packaging machine puts
in these bags is believed to have a normal
model with mean 10.2 ouches and
standard deviation 0.12 ounces. Describe
a sampling distribution if you consider 10
bags.
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What percent of the samples have an
average of 10 or less ounces?
Another Example
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The weight of adult males has a
mean of around 65 kg and a
standard deviation of 20 kg. Suppose
that a sample size of 16 is big
enough for the central limit theorem
to apply to the average weight of
males. What is the probability that
the average weight of 16 randomly
selected males will be over 75 kg?
Helpful Flowchart
What Am I Given?
Population Proportion
and Sample Size
Check
Conditions!
Calculate Sample’s
Standard Deviation
and Label Mean
Population
Mean, Standard
Deviation, and
Sample Size
Calculate Sample’s
Standard Deviation
Population Mean and
Standard Deviation
(no sample, just
individuals in a
population)
Z-SCORE PROBLEM
What is the problem asking for?
Probability
Z-SCORE
Describe Proportions
DRAW PICTURE AND
SENTENCES
Exit Ticket
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Exit Ticket and start homework