Transcript Chapter 7.1, 7.2

Estimates and
Sample Sizes
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1
But, first
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2
Example 1:
In a recent poll, 70% of 1501 randomly selected
adults said they believed in global warming.
Q: What is the proportion of the adult
population that believe in global warming?
Notation: p is the population proportion (an
unknown parameter).
pˆ is the sample proportion (computed). From
the poll data pˆ = 0.70.
Apparently, 0.70 will be the best estimate of the
proportion of all adults who believe in global
warming.
pg 329
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3
Definition
A point estimate is a single value (or
point) used to approximate a
population parameter.
pg 329
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4
Definition
The sample proportion pˆ is
the best point estimate of the
population proportion p.
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Example (continued)
We say that 0.70, or 70% is the best point estimate
of the proportion of all adults who believe in
global warming.
But how reliable (accurate) is this estimate?
We will see that its margin of error is 2.3%. This
means the true proportion of adults who believe in
global warming is between 67.7% and 72.3%. This
gives an interval (from 67.7% to 72.3%)
containing the true (but unknown) value of the
population proportion.
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What Is the Problem?
We have a point estimate, but we don’t
know how good it is.
We need to have a range of values.
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Definition
A confidence interval (or interval
estimate) is a range (or an interval) of
values used to estimate the true value of
a population parameter.
A confidence interval is sometimes
abbreviated as CI.
pg 329
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Notation for Critical Value
α = Greek letter alpha
Usually the confidence interval
is 1- α
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Definition
A confidence level is the probability 1 –  (often
expressed as the equivalent percentage value) that the
confidence interval actually does contain the population
parameter.
The confidence level is also called degree of confidence,
or the confidence coefficient.
Most common choices are 90%, 95%, or 99%.
( = 10%), ( = 5%), ( = 1%)
pg 330
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Example (continued)
In a recent poll, 70% of 1501 randomly selected
adults said they believed in global warming.
The sample proportion pˆ = 0.70 is the best estimate
of the population proportion p.
A 95% confidence interval for the unknown
population parameter is
0.677 < p < 0.723
What does it mean, exactly?
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Interpreting a Confidence Interval
Correct
We are 95% confident that the interval from 0.677
to 0.723 actually does contain the true value of the
population proportion p.
This means that if we were to select many different
samples of size 1501 and construct the
corresponding confidence intervals, then 95% of
them would actually contain the value of the
population proportion p.
pg 330
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12
Interpreting a Confidence Interval
Incorrect
There is a 95% chance that the true value of p will
fall between .0677 and .0723.
It would also be incorrect to say that 95% of sample
proportions fall between .0677 and .0723
pg 330
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Caution
Know the correct interpretation of a
confidence interval.
It is wrong to say
“the probability that the population
parameter belongs to the confidence
interval is 95%”
because the population parameter is not a
random variable, it does not change its
value.
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Caution
A confidence level of 95% tells us that the
process we are using will result in
confidence level limits that contain the true
population proportion 95% of the time.
We expect that 19 out of 20 samples should
contain confidence levels that contain the
true value of p
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Caution
Do not confuse two percentages: the
proportion may be represented by percents
(like 70% in the example), and the confidence
level may be represented by percents (like
95% in the example).
Proportion may be any number from 0% to
100%.
Confidence level is usually 90% or 95% or
99%.
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Confidence Interval for a
Population Proportion p
pˆ – E < p < pˆ + E
pˆ + E
(pˆ – E, pˆ+ E)
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Finding the Point Estimate and E
from a Confidence Interval
ˆ
(upper confidence limit) + (lower confidence limit)
Point estimate of p:
ˆ
p=
2
Margin of Error:
E = (upper confidence limit) — (lower confidence limit)
2
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Next we learn how to
construct confidence
intervals
pg 331
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Critical Values
A z score can be used to distinguish between sample
statistics that are likely to occur and those that are
unlikely to occur. Such a z score is called a critical
value.
The standard normal distribution is divided into three
regions: middle part has area 1-α and two tails
(left and right) have area α/2 each:
pg 331
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Critical Values
The z scores separate the middle interval (likely
values) from the tails (unlikely values). They are
zα/2 and – zα/2 , found from Table A-2.
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Definition
A critical value is the number on the
borderline separating sample
statistics that are likely to occur
from those that are unlikely to occur.
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Notation for Critical Value
The critical value z/2 separates an area
of /2 in the right tail of the standard
normal distribution. The value of –z/2
separates an area of /2 in the left tail.
The subscript /2 is simply a reminder
that the z score separates an area of /2
in the tail.
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Finding zα/2
Confidence
Level
α
zα/2
90%
95%
99%
0.10
0.05
0.01
1.645
1.96
2.575
pg 332
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Finding zα/2 for a 95% Confidence
Level
α = 5%
α/ 2 = 2.5% = .025
- zα/2
Critical Values
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zα/2
pg 332
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Definition
Margin of error, denoted by E, is the maximum likely
difference (with probability 1 – , such as 0.95)
between the observed proportion pˆ and the true value
of the population proportion p.
The margin of error E is also called the maximum
error of the estimate.
pg 333
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Margin of Error for
Proportions
E  z 2
ˆˆ
pq
n
Formula 7-1, pg 333
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Notation
E = margin of error
p^ = sample proportion
q^ = 1 – ^
p
n = number of sample values
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Confidence Interval for a
Population Proportion p
pˆ – E < p < pˆ+ E
where
E  z 2
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ˆˆ
pq
n
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Confidence Interval for a
Population Proportion p
pˆ – E < p < pˆ + E
Also written as:
pˆ + E
Also written as:
ˆ – E, pˆ + E)
(p
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Finding the Point Estimate and E
from a Confidence Interval
ˆ
(upper confidence limit) + (lower confidence limit)
Point estimate of p:
ˆ
p=
2
Margin of Error:
E = (upper confidence limit) — (lower confidence limit)
2
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31
Round-Off Rule for
Confidence Interval Estimates of p
Round the confidence interval limits for
p to
three significant digits.
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Example 3:
In a recent poll, 70% of 1501 randomly selected
adults said they believed in global warming.
a. Find the margin of error E that corresponds
to a 95% confidence level.
b. Find the 95% confidence interval estimate of
the population proportion p.
c. Based on the results, can we safely conclude
that the majority of adults believe in global
warming?
d. Assuming that you are a newspaper reporter,
write a brief statement that describes the
results accurately, with all relevant
pg 334
information.
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Example 3:
In a recent poll, 70% of 1501 randomly selected
adults said they believed in global warming.
a.Find the margin of error E that corresponds to
a 95% confidence level.
Zα/2 = 1.96 (Slide 24)
p=.70
q=.30
n=1501
E = 1.96 * sqrt(.70*.30/1501) = .023183 Slide 27
pg 334
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Example 3:
In a recent poll, 70% of 1501 randomly selected
adults said they believed in global warming.
b. Find the 95% confidence interval estimate of
the population proportion p.
pˆ – E < p < pˆ + E
.70 - .023183 < p < .70 + .023183
.677 < p < .723
pg 334
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Example 3:
In a recent poll, 70% of 1501 randomly selected
adults said they believed in global warming.
c. Based on the results, can we safely conclude
that the majority of adults believe in global
warming?
Based on the confidence interval obtained in b. it
appears that the proportion of adults who
believe in global warming is greater than .5,
so we can conclude that the majority of adults
believe in global warming.
pg 334
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Example 3:
In a recent poll, 70% of 1501 randomly selected
adults said they believed in global warming.
c. Based on the results, can we safely conclude
that the majority of adults believe in global
warming?
Yes.
pg 334
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Example 3:
In a recent poll, 70% of 1501 randomly selected
adults said they believed in global warming.
d.Assuming that you are a newspaper reporter,
write a brief statement that describes the results
accurately, with all relevant information.
70% of U.S. adults believe that the earth is getting
warmer, based on a Pew Research Center poll of 1501
randomly selected adults in the US. In theory, in 95% of
such polls, the percentage should differ by no more than
2.3% in either direction from the percentage that would
be found by interviewing all adults in the U.S.
pg 334
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Confidence Intervals by TI-83/84
•
•
•
•
•
•
•
•
•
Press STAT and select TESTS
Scroll down to 1-PropZInt
and press ENTER
Type in x: (number of successes)
n: (number of trials)
C-Level: (confidence level)
Press on Calculate
Read the confidence interval (…..,..…)
^
and the point estimate p=…
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Confidence Intervals by Excel
Example 3, pg 334.
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Enter data
40
Confidence Intervals by Excel
Enter formula
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Confidence Intervals by Excel
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Sample Size
pg 336
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Sample Size
Suppose we want to collect sample data in
order to estimate some population
proportion. The question is how many
sample items must be obtained?
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Determining Sample Size
(solve for n by algebra)
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Sample Size for Estimating
Proportion p
ˆ
When an estimate of p is known:
ˆ
When no estimate of p is known:
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Round-Off Rule for Determining
Sample Size
If the computed sample size n is not a
whole number, round the value of n up to
the next larger whole number.
Examples:
n=310.67 round up to 311
n=310.23 round up to 311
n=310.01 round up to 311
pg 336
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Example:
A manager for E-Bay wants to determine the
current percentage of U.S. adults who now use the
Internet.
How many adults must be surveyed in order to be
95% confident that the sample percentage is in
error by no more than three percentage points?
a) In 2006, 73% of adults used the Internet.
b) No known possible value of the proportion.
Example 4, pg 336
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Example:
a) Use
pˆ  0.73 and qˆ  1  pˆ  0.27
  0.05 so z 2  1.96
E  0.03
To be 95% confident
that our sample
percentage is within
three percentage points
of the true percentage
for all adults, we should
obtain a random sample
of 842 adults.
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Example:
b) Use
a = 0.05 so za 2 = 1.96
E = 0.03
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This time we don’t know
the sample proportion
To be 95% confident that
our sample percentage is
within three percentage
points of the true
percentage for all adults, we
should obtain a random
sample of 1068 adults.
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