Transcript Document

Chapter 16
Inference about a Population Mean
BPS - 3rd Ed.
Chapter 16
1
Conditions for Inference
about a Mean


Data are a SRS of size n
Population has a Normal distribution
with mean m and standard deviation s
– Both m and s are unknown

Because s is unknown, we cannot
use z procedures to infer µ
– This is a more realistic situation than in
prior chapters
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Standard Error

When we do not know population standard
deviation s, we use sample standard deviation
s to calculate the standard deviation of x-bar,
which is now equal to
s
n
This statistic is called the
standard error of the mean

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One-Sample t Statistic

When we use s instead of s,our “one-sample
z statistic” becomes a one-sample t statistic
x  μ0
z
σ
n


x  μ0
t
s
n
The t test statistic does NOT follow a Normal
distribution  it follows a t distribution with
n – 1 degrees of freedom
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The t Distributions

The t density curve is similar to the standard
Normal curve (symmetrical, mean = 0, bellshaped), but has broader tails

The t distribution is a family of distributions
–
Each family member is identified by its degree
of freedom
–
Notation t(k) means “a t distribution with k
degrees of freedom”
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The t Distributions
As k increases,
the t(k) density
curve approaches
the Z curve more
closely.
This is because s
becomes a better
estimate of s as
the sample size
increases
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Using Table C
Table C gives t critical values having upper tail probability
p along with corresponding confidence level C
The bottom row of the table applies to z* because a t* with infinite degrees of
freedom is a standard Normal (Z) variable
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Using Table C
This t table below highlights the t* critical value with
probability 0.025 to its right for the t(7) curve
t* = 2.365 (for 7 df)
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One-Sample t Confidence Interval
Take a SRS of size n from a population with unknown mean
m and unknown standard deviation s.
A level C confidence interval for m is given by:
x t

s
n
where t* is the critical value for confidence level C from the
t(n – 1) density curve.
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Illustrative example
American Adult Heights
A study of 8 American adults from a SRS yields an
average height of x = 67.2 inches and a standard
deviation of s = 3.9 inches. A 95% confidence interval
for the average height of all American adults (m) is:
x t


s
n
 67.2  2.365
3.9
 67.2  3.3
8
 63.9 to 70.5 inches
“We are 95% confident that the average height of all
American adults is between 63.9 and 70.5 inches.”
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One-Sample t Test
The t test is similar in form to the z test learned
earlier. The test statistic is:
x  μ0
t
s n
The P-value of t is determined from the t(n – 1)
curve via the t table.
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P-value for Testing Means

Ha: m> m0


Ha: m< m0


P-value is the probability of getting a value as large or
larger than the observed test statistic (t) value.
P-value is the probability of getting a value as small or
smaller than the observed test statistic (t) value.
Ha: mm0

P-value is two times the probability of getting a value as
large or larger than the absolute value of the observed test
statistic (t) value.
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BPS - 3rd Ed.
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Illustrative example
Weight Gain
To study weight gain, ten people between the age of 30
and 40 determine their change in weight over a 1-year
interval. Here are their weight changes:
2.0
0.4
0.7
2.0
-0.4
2.2
-1.3
1.2
1.1
2.3
Are these data statistically significant evidence that
people in this age range gain weight over a given year?
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Weight Gain Illustration
1.
2.
3.
4.
Hypotheses:
Test Statistic: t 
(df = 101 = 9)
H 0: m = 0
H a: m > 0
x  μ0
s

1.02  0
 2.70
1.196
n
10
P-value:
P-value = P(T > 2.70) = 0.0123 (using a computer)
P-value is between 0.01 and 0.02 since t = 2.70 is between
t* = 2.398 (p = 0.02) and t* = 2.821 (p = 0.01) (Table C)
Conclusion:
Since the P-value is smaller than a = 0.05, there is significant
evidence that against the null hypothesis.
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Illustrative example
Weight Gain
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Matched Pairs t Procedures



Matched pair samples allow us to
compare responses to two treatments in
paired couples
Apply the one-sample t procedures to
the observed differences within pairs
The parameter m is the mean difference
in the responses to the two treatments
within matched pairs in the entire
population
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Case Study – Matched Pairs
Air Pollution



Pollution index
measurements were
recorded for two areas of a
city on each of 8 days
To analyze, subtract Area B
levels from Area A levels.
The 8 differences form a
single sample.
Are the average pollution
levels the same for the two
areas of the city?
Day
Area A Area B
A–B
1
2.92
1.84
1.08
2
1.88
0.95
0.93
3
5.35
4.26
1.09
4
3.81
3.18
0.63
5
4.69
3.44
1.25
6
4.86
3.69
1.17
7
5.81
4.95
0.86
8
5.55
4.47
1.08
These 8 differences have x = 1.0113 and s = 0.1960.
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Case Study
1.
Hypotheses:
2.
Test Statistic:
(df = 81 = 7)
H 0: m = 0
H a: m ≠ 0
t 
x  μ0
s

1.0113  0
0.1960
n
3.
4.
 14.594
8
P-value:
P-value = 2P(T > 14.594) = 0.0000017 (using a computer)
P-value is smaller than 2(0.0005) = 0.0010 since t = 14.594 is
greater than t* = 5.408 (upper tail area = 0.0005) (Table C)
Conclusion:
Since the P-value is smaller than a = 0.001, there is strong
evidence (“highly signifcant”) that the mean pollution levels are
different for the two areas of the city.
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Case Study
Air Pollution
Find a 95% confidence interval to estimate the
difference in pollution indexes (A – B) between the two
areas of the city. (df = 81 = 7 for t*)
0.1960
 s
x t
 1.0113  2.365
 1.0113  0.1639
n
8
 0.8474 to 1.1752
We are 95% confident that the pollution index in area
A exceeds that of area B by an average of 0.8474 to
1.1752 index points.
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Conditions for t procedures
Weight Gain Illustration
1.
2.
3.
4.
Valid data?
Unconfounded comparison? (lurking
variables are balanced)
Simple random sample?
Normal distribution?
Except in the case of small samples, conditions 1 –3 are
of more importance than condition 4 (t procedures are
“robust” to violations in condition 4).
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Robustness of Normality Assumption



The t confidence interval and test are exact
when the distribution of the population is
Normal
However, no real data are exactly Normal
A confidence interval or significance test is
robust when the confidence level or P-value
does not change very much when the
conditions for use of the procedure are violated
t statistics are robust when n is
moderate to large
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t Procedure Robustness



Sample size less than 15: Use t procedures if the data
appear about Normal (symmetric, single peak, no outliers).
If the data are skewed or if outliers are present, do not use
t.
Sample size at least 15: The t procedures can be used
except in the presence of outliers or strong skewness in
the data.
Large samples: The t procedures can be used even for
clearly skewed distributions when the sample is large,
roughly n ≥ 40.
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Assessing Normality
Air Pollution Illustration




Recall the air pollution illustration
Is it reasonable to assume the
sampling distribution of xbar is
Normal?
While we cannot judge Normality
from just 8 observations, a stemplot
of the data (right) shows no outliers,
clusters, or extreme skewness.
Thus, the t test will be accurate.
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Can we use t?
The stemplot shows a moderately sized data set (n = 20)
with a strong negative skew.
We cannot trust t procedures in this instance
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Can we use t?


This histogram shows the distribution of word lengths
in Shakespeare’s plays. The sample is very large.
The data has a strong positive skew, but there are no
outliers. We can use the t procedures since n ≥ 40
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Can we use t?


This histogram shows the heights of college students.
The distribution is close to Normal, so we can trust the
t procedures for any sample size.
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