Transcript CHAPTER 17

CHAPTER 10
• CONFIDENCE INTERVALS FOR
ONE SAMPLE POPULATION
• CONFIDENCE INTERVAL FOR THE
DIFFERENCE IN TWO SAMPLE
POPULATION
1
Point Estimate and Interval
Estimate
A point estimate is a single number that is our
“best guess” for the parameter. Point
estimation produces a number (an estimate)
which is believed to be close to the value of the
unknown parameter.
An interval estimate is an interval of numbers
within which the parameter value is believed to
fall. Interval estimation produces an interval
that contains the estimated parameter with a
prescribed confidence.
2
Point Estimate and Interval Estimate
(Figure 10.1)
3
Point Estimate and Interval Estimate
• Figure 10.1 A point estimate predicts a
parameter by a single number. An
interval estimate is an interval of
numbers that are believable values for
the parameter.
• Question: Why is a point estimate alone
not sufficiently informative?
4
Point Estimate and Interval Estimate
A point estimate doesn’t tell us how
close the estimate is likely to be to the
parameter.
An interval estimate is more useful, it
incorporates a margin of error which
helps us to gauge the accuracy of the
point estimate.
5
Properties of Point Estimators
Property 1: A good estimator has a sampling
distribution that is centered at the parameter.
An estimator with this property is unbiased.
 The sample mean is an unbiased estimator
of the population mean.
 The sample proportion is an unbiased
estimator of the population proportion.
6
SOME POINT ESTIMATORS
PARAMETER
PROPORTION
MEAN
STANDARD
DEVIATION
P


UNBIASED
ESTIMATOR
Pˆ
X
S
7
Properties of Point Estimators
Property 2: A good estimator has a small standard
deviation compared to other estimators.
 This means it tends to fall closer than other
estimates to the parameter.
 The sample mean has a smaller standard error than
the sample median when estimating the population
mean of a normal distribution.
8
The Logic behind Constructing a
Confidence Interval
To construct a confidence interval for a
population proportion, start with the sampling
distribution of a sample proportion.
 Gives the possible values for the sample
proportion and their probabilities.
The sampling distribution:
 Is approximately a normal distribution for
large random samples by the CLT.
 Has mean equal to the population proportion.
 Has standard deviation called the standard
error.
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Constructing a Confidence Interval to Estimate
a Population Proportion
 We symbolize a population proportion
by p.
 The point estimate of the population
proportion is the sample proportion.
 We symbolize the sample proportion by
ˆ
p
called “p-hat”.
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Constructing a Confidence Interval to Estimate
a Population Proportion
• A CONFIDENCE INTERVAL OFTEN HAS THE
FORM:
POINT ESTIMATE MARGIN OF ERROR(ME)
• IT IS CONSTRUCTED WITH A PRESCRIBED
CONFIDENCE KNOWN AS THE CONFIDENCE
LEVEL
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Confidence Interval or Interval Estimate
Sample estimate  Multiplier × Standard Error
Sample estimate  Margin of error
• Multiplier is a number based on the confidence
level desired and determined from the standard
normal distribution (for proportions) or Student’s
t-distribution (for means).
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The Multiplier
• Multiplier, denoted as z*, is the
standardized score such that the area
between -z* and z* under the standard
normal curve corresponds to the
desired confidence level.
• Note: Increase confidence level =>
larger multiplier
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The Multiplier
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For 90% Confidence Level
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SOME CRITICAL VALUES FOR STANDARD
NORMAL DISTRIBUTION
C % CONFIDENCE
LEVEL
80%
CRITICAL VALUE
90%
1.645
95%
1.960
98%
2.326
99%
2.576
Z*
1.282
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Interpretation of the Confidence
Level
So what does it mean to say that we have “95%
confidence”?
The meaning refers to a long-run interpretation—how
the method performs when used over and over with
many different random samples.
If we used the 95% confidence interval method over
time to estimate many population proportions, then in
the long run about 95% of those intervals would give
correct results, containing the population proportion.
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WHAT DOES C% CONFIDENCE REALLY
MEAN?
• FORMALLY, WHAT WE MEAN IS THAT C% OF
SAMPLES OF THIS SIZE WILL PRODUCE
CONFIDENCE INTERVALS THAT CAPTURE THE
TRUE PROPORTION.
• C% CONFIDENCE MEANS THAT ON AVERAGE, IN C
OUT OF 100 ESTIMATIONS, THE INTERVAL WILL
CONTAIN THE TRUE ESTIMATED PARAMETER.
• E.G. A 95% CONFIDENCE MEANS THAT ON THE
AVERAGE, IN 95 OUT OF 100 ESTIMATIONS, THE
INTERVAL WILL CONTAIN THE TRUE ESTIMATED
PARAMETER.
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CONFIDENCE INTERVAL FOR PROPORTION P
[ONE-PROPORTION Z-INTERVAL]
•
ASSUMPTIONS AND CONDITIONS
RANDOMIZATION CONDITION
•
10% CONDITION
•
SAMPLE SIZE ASSUMPTION OR
SUCCESS/FAILURE CONDITION
•
•
INDEPENDENCE ASSUMPTION
NOTE: PROPER RANDOMIZATION CAN HELP
ENSURE INDEPENDENCE.
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CONSTRUCTING CONFIDENCE
INTERVALS
ESTIMATOR
STANDARD ERROR
C% MARGIN OF
ERROR
C% CONFIDENCE
INTERVAL
SAMPLE PROPORTION
ˆ
P
SE( Pˆ ) 
pˆ qˆ
n
ME( pˆ )  z SE( pˆ )
*
pˆ  ME( pˆ )
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Compact Formula For a Confidence
Interval For a Population Proportion p
pˆ  z

pˆ 1  pˆ 
n
ˆ is the sample proportion.
• p
• z* denotes the multiplier.
where
•
ˆ 1  p
ˆ
p
n
is the standard error of pˆ .
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Constructing a Confidence Interval to Estimate
a Population Proportion
•The exact standard deviation of a sample
proportion equals: p(1  p)
n
•This formula depends on the unknown
population proportion, p.
•In practice, we don’t know p, and we need
to estimate the standard error as
se 
ˆ (1  p
ˆ)
p
n
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Margin of Error
– The margin of error measures how accurate
the point estimate is likely to be in estimating a
parameter.
– It is a multiple of the standard error of the
sampling distribution when the sampling
distribution is a normal distribution.
– The distance of 1.96 standard errors is the
margin of error for a 95% confidence interval
for a parameter from a normal distribution.
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Intuitive Explanation of Margin of Error
• Margin of Error Characteristics:
• The difference between the sample proportion and
the population proportion is less than the margin of
error about 95% of the time, or for about 19 of every
20 sample estimates.
• The difference between the sample proportion and
the population proportion is more than the margin of
error about 5% of the time, or for about 1 of every 20
sample estimates
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SAMPLE SIZE NEEDED TO PRODUCE A CONFIDENCE
INTERVAL WITH A GIVEN MARGIN OF ERROR, ME
ˆ)  z
ME( p
SOLVING FOR n GIVES
*
ˆ qˆ
p
n
ˆ qˆ
(z ) p
n
2
( ME )
* 2
ˆ AND qˆ IS A REASONABLE GUESS. IF WE
WHERE p
CANNOT MAKE A GUESS, WE TAKE p
ˆ  qˆ  0.5
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EXAMPLE 1
A MAY 2002 GALLUP POLL FOUND THAT ONLY 8% OF A
RANDOM SAMPLE OF 1012 ADULTS APPROVED OF
ATTEMPTS TO CLONE A HUMAN.
(A)
(B)
(C)
(D)
(E)
FIND THE MARGIN OF ERROR FOR THIS POLL IF WE WANT
95% CONFIDENCE IN OUR ESTIMATE OF THE PERCENT OF
AMERICAN ADULTS WHO APPROVE OF CLONING HUMANS.
EXPLAIN WHAT THAT MARGIN OF ERROR MEANS.
IF WE ONLY NEED TO BE 90% CONFIDENT, WILL THE
MARGIN OF ERROR BE LARGER OR SMALLER? EXPLAIN.
FIND THAT MARGIN OF ERROR.
IN GENERAL, IF ALL OTHER ASPECTS OF THE SITUATION
REMAIN THE SAME, WOULD SMALLER SAMPLES PRODUCE
SMALLER OR LARGER MARGINS OF ERROR?
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SOLUTION
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EXAMPLE 2
DIRECT MAIL ADVERTISERS SEND SOLICITATIONS (a.k.a. “junk
mail”) TO THOUSANDS OF POTENTIAL CUSTOMERS IN THE
HOPE THAT SOME WILL BUY THE COMPANY’S PRODUCT.
THE RESPONSE RATE IS USUALLY QUITE LOW. SUPPOSE
A COMPANY WANTS TO TEST THE RESPONSE TO A NEW
FLYER, AND SENDS IT TO 1000 PEOPLE RANDOMLY
SELECTED FROM THEIR MAILING LIST OF OVER 200,000
PEOPLE. THEY GET ORDERS FROM 123 OF THE
RECIPIENTS.
(A) CREATE A 90% CONFIDENCE INTERVAL FOR THE
PERCENTAGE OF PEOPLE THE COMPANY CONTACTS WHO
MAY BUY SOMETHING.
(B) EXPLAIN WHAT THIS INTERVAL MEANS.
(C) EXPLAIN WHAT “90% CONFIDENCE” MEANS.
(D) THE COMPANY MUST DECIDE WHETHER TO NOW DO A
MASS MAILING. THE MAILING WON’T BE COST-EFFECTIVE
UNLESS IT PRODUCES AT LEAST A 5% RETURN. WHAT
DOES YOUR CONFIDENCE INTERVAL SUGGEST? EXPLAIN.
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SOLUTION
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EXAMPLE 3
IN 1998 A SAN DIEGO REPRODUCTIVE CLINIC
REPORTED 49 BIRTHS TO 207 WOMEN UNDER
THE AGE OF 40 WHO HAD PREVIOUSLY BEEN
UNABLE TO CONCEIVE.
(A) FIND A 90% CONFIDENCE INTERVAL FOR THE
SUCCESS RATE AT THIS CLINIC.
(B) INTERPRET YOUR INTERVAL IN THIS CONTEXT.
(C) EXPLAIN WHAT “90 CONFIDENCE” MEANS.
(D) WOULD IT BE MISLEADING FOR THE CLINIC TO
ADVERTISE A 25% SUCCESS RATE? EXPLAIN.
(E) THE CLINIC WANTS TO CUT THE STATED
MARGIN OF ERROR IN HALF. HOW MANY
PATIENTS’ RESULTS MUST BE USED?
(F) DO YOU HAVE ANY CONCERNS ABOUT THIS
SAMPLE? EXPLAIN.
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SOLUTION
31
How Can We Use Confidence Levels
Other than 95%?
In practice, the confidence level 0.95 is
the most common choice. But, some
applications require greater (or less)
confidence.
•
To increase the chance of a correct
inference, we can use a larger confidence
level, such as 0.99.
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A 99% Confidence Interval Is Wider Than a 95%
Confidence Interval.
33
Question: If you want greater confidence, why
would you expect a wider interval?
• In using confidence intervals, we must
compromise between the desired margin
of error and the desired confidence of a
correct inference.
– As the desired confidence level
increases, the margin of error gets larger.
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Effects of Confidence Level and Sample Size
on Margin of Error
The margin of error for a confidence interval:
 Increases as the confidence level increases
 Decreases as the sample size increases
For instance, a 99% confidence interval is wider than
a 95% confidence interval, and a confidence interval
with 200 observations is narrower than one with 100
observations at the same confidence level. These
properties apply to all confidence intervals, not just the
one for the population proportion.
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What is the Error Probability for the
Confidence Interval Method?
•The general formula for the confidence
interval for a population proportion is:
•
Sample estimate  Multiplier × Standard Error
–which in symbols is
pˆ  z(se)
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What is the Error Probability for the
Confidence Interval Method?
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Confidence Intervals for the Difference
Between Two Proportions
p1  p2  z
*
p1 1  p1  p2 1  p2 

n1
n2
where z* is the value of the standard
normal variable with area between -z*
and z* equal to the desired confidence
level.
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Necessary Conditions
• Condition 1: Sample proportions are
available based on independent, randomly
selected samples from the two populations.
• Condition 2: All of the quantities –
n1 pˆ1, n1 1  pˆ1 , n2 pˆ 2 , and n2 1  pˆ 2 
– are at least 10.
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Example: Age and Using the Internet
Young:92 of 262 use Internet as main news source
ˆ1= .351
p
ˆ 2=
Old: 59 of 632 use Internet as main news source p
.093
pˆ1  pˆ 2  .351 .093 .258 and s.e.( pˆ1  pˆ 2 )  .0317
• Approximate 95% Confidence Interval:
.258  1.96(.0317)  .196 to .320
• We are 95% confident that somewhere between
19.6% and 32.0% more young adults than older
adults use the Internet as their main news source.
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Using Confidence Intervals to Guide
Decisions
Principle 1. A value not in a confidence interval can be
rejected as a possible value of the population
proportion. A value in a confidence interval is an
“acceptable” possibility for the value of a population
proportion.
Principle 2. When a confidence interval for the
difference in two population proportions does not
cover 0, it is reasonable to conclude the two population
proportions are different.
Principle 3. When the confidence intervals for
proportions in two different populations do not
overlap, it is reasonable to conclude the two
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population proportions are different.
Example: Which Drink Tastes Better?
• Taste Test: A sample of 60 people taste both drinks
and 55% like taste of Drink A better than Drink B
Makers of Drink A want to advertise these results.
Makers of Drink B make a 95% confidence interval for
the population proportion who prefer Drink A.
95% Confidence Interval:
.551  .55
.55  2
 .55  .13
60
• Note: Since .50 is in the interval, there is not enough
evidence to claim that Drink A is preferred by a
majority of population represented by the sample.
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CHAPTER 11
ESTIMATING MEANS WITH
CONFIDENCE
43
CONFIDENCE INTERVALS FOR ONE
POPULATION MEAN
The confidence interval again has the form
Point estimate  margin of error
The sample mean is the point estimate of the
population mean.
The exact standard error of the sample mean is
/ n
• In practice, we estimate σ by the sample standard
• deviation, s, so s.e. x   s
n
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Confidence Intervals for One
Population Mean
• For large n… from any population and also
• For small n from an underlying population
that is normal…
• The confidence interval for the population
mean is:
x  z(

n
)
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Confidence Intervals for One
Population Mean
In practice, we don’t know the population standard
deviation .
•
Substituting the sample standard deviation s for to
s
get
s.e. x  
n
•
introduces extra error. To account for this
increased error, we must replace the z-score by a
slightly larger score, called a t –score. The confidence
interval is then a bit wider. This distribution is called
the t distribution.

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Summary: Properties of the t-Distribution
 The t-distribution is bell shaped and symmetric
about 0.
 The probabilities depend on the degrees of freedom,
df  n  1.
 The t-distribution has thicker tails than the standard
normal distribution, i.e., it is more spread out.
 A t -score multiplied by the standard error gives the
margin of error for a confidence interval for the
mean.
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t - Distribution
48
t - Distribution
• The t Distribution Relative to the Standard
Normal Distribution: The t distribution gets
closer to the standard normal as the degrees
of freedom ( df ) increase. The two are
practically identical when df  30 .
• Question: Can you find z -scores (such as
1.96) for a normal distribution on the t table?
49
t - Distribution
50
t – Distribution
• Part of t - Table Displaying t-Scores. The
scores have right-tail probabilities of 0.100,
0.050, 0.025, 0.010, 0.005, and 0.001. When
n  7, df  6
• and t.025  2.447 is the t -score with right-tail
probability = 0.025 and two-tail probability =
0.05. It is used in a 95% confidence interval,
x  2.447( se)
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t - Distribution
52
t - Distribution
• The t Distribution with df = 6. 95% of the
distribution falls between -2.447
• and 2.447. These t -scores are used with a
95% confidence interval when n = 7.
• Question: Which t -scores with df = 6 contain
the middle 99% of a t distribution (for a 99%
confidence interval)?
53
Using the t Distribution to Construct a
Confidence Interval for a Mean
•Summary: 95% Confidence Interval for a Population Mean
•When the standard deviation of the population is unknown, a 95%
confidence interval for the population mean  is:
s
x
t( );
df

n
1
n
.025
•To use this method, you need:
 Data obtained by randomization
 An approximately normal population distribution
54
SUMMARY
55
ASSUMPTIONS AND CONDITIONS
• INDEPENDENCE ASSUMPTION: THE DATA VALUES
SHOULD BE INDEPENDENT. THERE’S REALLY NO
WAY TO CHECK INDEPENDENCE OF THE DATA BY
LOOKING AT THE SAMPLE, BUT WE SHOULD
THINK ABOUT WHETHER THE ASSUMPTION IS
REASONABLE.
• RANDOMIZATION CONDITION: THE DATA SHOULD
ARISE FROM A RANDOM SAMPLE OR SUITABLY A
RANDOMIZED EXPERIMENT.
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ASSUMPTIONS AND CONDITIONS
• 10% CONDITION: THE SAMPLE IS NO MORE
THAN 10% OF THE POPULATION.
• NORMAL POPULATION ASSUMPTION OR
NEARLY NORMAL CONDITION: THE DATA
COME FROM A DISTRIBUTION THAT IS
UNIMODAL AND SYMMETRIC. REMARK:
CHECK THIS CONDITION BY MAKING A
HISTOGRAM OR NORMAL PROBABILITY
PLOT.
57
CONSTRUCTING CONFIDENCE INTERVALS
FOR MEANS
• POINT ESTIMATOR:
• STANDARD ERROR:
• C% MARGIN OF ERROR:
58
WHERE tn-1* IS A CRITICAL VALUE FOR STUDENT’S t
– MODEL WITH n – 1 DEGREES OF FREEDOM THAT
CORRESPONDS TO C% CONFIDENCE LEVEL.
n
(t
*
n1
2
) s
2
ME
2
59
REMARK
60
ILLUSTRATIVE PICTURE
61
FINDING CRITICAL t - VALUES
• Using t tables (Table T) and/or calculator, find or
estimate the
• 1. critical value t7* for 90% confidence level if
number of degrees of freedom is 7
• 2. one tail probability if t = 2.56 and number of
degrees of freedom is 7
• 3. two tail probability if t = 2.56 and number of
degrees of freedom is 7
• NOTE: If t has a Student's t-distribution with
degrees of freedom, df, then TI-83 function
tcdf(a,b,df) , computes the area under the t-curve
and between a and b.
62
EXAMPLES FROM MIDTERM EXAM III
PRACTICE EXERCISES
63
Choosing the Sample Size for Estimating
a Population Mean
In practice, you don’t know the value of the standard
deviation, .
•
You must substitute an educated guess for  .
•
Sometimes you can use the sample standard
deviation from a similar study.
When no prior information is known, a crude
estimate that can be used is to divide the estimated
range of the data by 6 since for a bell-shaped
distribution we expect almost all of the data to fall
within 3 standard deviations of the mean.
64
Other Factors That Affect the Choice of
the Sample Size
 The first is the desired precision, as
measured by the margin of error, m.
 The second is the confidence level.
 The third factor is the variability in the data.
 The fourth factor is cost.
65
What if You Have to Use a Small n?
The t methods for a mean are valid for any n.
However, you need to be extra cautious to
look for extreme outliers or great departures
from the normal population assumption.
– In the case of the confidence interval for a
population proportion, the method works
poorly for small samples because the CLT no
longer holds.
66
Confidence Intervals for Difference in Two
Population Means (Independent Samples)
67
Confidence Intervals for Difference for the
Difference Between Two Population Means
Approximate CI for 1 – 2:
x1  x2  t *
s12 s22

n1 n2
where t* is the value in a t-distribution with area
between -t* and t* equal to the desired confidence
level.
Approximate df difficult to specify.
Use computer software or conservatively use the
smaller of the two sample sizes and subtract 1.
68
Degrees of Freedom
The t-distribution is only approximately
correct and df formula is complicated
(Welch’s approximation):
Statistical software can use the above
approximation, but if done by-hand then use a
conservative df = smaller of n1 – 1 and n2 – 1.
69
Necessary Conditions
Two samples must be independent and either:
Situation 1: Populations of measurements both
bell-shaped, and random samples of any size
are measured.
Situation 2: Large (n  30) random samples are
measured. But if there are extreme outliers,
or extreme skewness, it is better to have an
even larger sample than n = 30.
70
Example: Effect of a Stare on Driving
• Randomized experiment: Researchers either stared
or did not stare at drivers stopped at a campus stop
sign; Timed how long (sec) it took driver to proceed
from sign to a mark on other side of the
intersection.
No Stare Group (n = 14): 8.3, 5.5, 6.0, 8.1, 8.8, 7.5, 7.8,
7.1, 5.7, 6.5, 4.7, 6.9, 5.2, 4.7
Stare Group (n = 13):
5.6, 5.0, 5.7, 6.3, 6.5, 5.8, 4.5,
6.1, 4.8, 4.9, 4.5, 7.2, 5.8
• Task:
Make a 95% CI for the difference between
the mean crossing times for the two populations
represented by these two independent samples.
71
Example: Effect of a Stare on Driving
72
Example: Effect of a Stare on Driving
Checking Conditions
Boxplots show …
• No outliers and no strong skewness.
• Crossing times in stare group generally
faster and less variable.
73
Example: Effect on a Stare on Driving
Note: The df = 21 was reported by the computer package
based on the Welch’s approximation formula.
74
Equal Variance Assumption and the
Pooled Standard Error
• May be reasonable to assume the two populations
have equal population standard deviations, or
2
2
2





equivalently, equal population variances: 1
2
• Estimate of this variance based on the combined
or “pooled” data is called the pooled variance.
The square root of the pooled variance is called
the pooled standard deviation:
Pooled standard deviation s p 
n1  1s12  n2  1s22
n1  n2  2
75
Pooled Standard Error
P ooleds.e.( x1  x2 ) 

s
2
p
n1

s
2
p
n2
1
1
s   
 n1 n2 
 sp
2
p
1
1

n1 n2
76
Pooled Degrees of Freedom
(df)
• Note: Pooled df = (n1 – 1) + (n2 – 1)
= (n1 + n2 – 2).
77
Pooled Confidence Interval
Pooled CI for the Difference Between
Two Means (Independent Samples):


1
1
x1  x2  t  s p
 
n
n
1
2 

*
where t* is found using a t-distribution
with df = (n1 + n2 – 2) and
sp is the pooled standard deviation.
78
Example: Male and Female Sleep
Times
• Q: How much difference is there between how long
female and male students slept the previous night?
• Data:
The 83 female and 65 male responses from
students in an intro stat class.
• Task:
Make a 95% CI for the difference between
the two population means sleep hours for females
versus males.
• Note: We will assume equal population variances.
79
Example: Male and Female Sleep
Times
Two-sample T for sleep [with “Assume Equal Variance”
option]
Sex
N
Mean StDev SE Mean
Female 83
7.02
1.75
0.19
Male
6.55
1.68
0.21
65
Difference = mu (Female) – mu (Male)
Estimate for difference: 0.461
95% CI for difference: (-0.103, 1.025)
T-Test of difference = 0 (vs not =): T-Value = 1.62 P =
0.108 DF = 146
Both use Pooled StDev = 1.72
80
Example: Male and Female Sleep
Times
Notes:
• Two sample standard deviations are
very similar.
• Sample mean for females higher than
for males.
• 95% confidence interval contains 0 so
cannot rule out that the population
means may be equal.
81
Example: Male and Female Sleep Times
• Pooled Standard Deviation and Pooled Standard
Error “by – hand”:
P ooledst d dev s p 

n1  1s12  n2  1s22
n1  n2  2
83  11.752  65  11.682
83  65  2
 2.957  1.72
82
Example: Male and Female Sleep
Times
P ooleds.e.( x1  x2 )  s p
1 1

n1 n2
1
1
 1.72

 0.285
83 65
83
Pooled or Unpooled?
• If the larger sample size produced the larger
standard deviation, the pooled procedure is
acceptable because it will be conservative.
• If the smaller standard deviation accompanies the
larger sample size, the pooled test can be quite
misleading and not recommended.
• If sample sizes are equal, the pooled and unpooled
standard errors are equal. Unless the sample
standard deviations are quite similar, it is best to use
the unpooled procedure.
84
Confidence Interval for the Difference in
Two Population Means
x1  x2   t
*
 s.e.x1  x2 
1.Make sure appropriate conditions apply checking
sample size and/or a shape picture of the differences.
2.Choose a confidence level.
3.Compute the mean and std dev for each sample.
4.Determine whether the std devs are similar enough to
pooled procedure can be used.
5.Calculate the appropriate standard error (pooled or
unpooled).
6.Calculate the appropriate df.
7.Use Table A.2 (or software) to find the multiplier t*.
85
Examples From Midterm Exam III
Practice Sheet
86