# Confidence Interval Review

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```					      Chapter 7

Inference for Distributions

1
• The previous chapter emphasized the reasoning
of tests and confidence intervals
• Now we emphasize statistical practice
– we no longer assume that population standard
deviations are known

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Section 7.1

Inference for the Mean of a
Population

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Introduction

• So far, in all our inference for the population mean ,
we have assumed that we know .
• The sampling distribution of x-bar depends on 
– Thus, both CI’s and significance tests depend on 
          x  0
x  (z*)       ,   z
n          / n

• But usually we don’t know . Then what?
– A sensible idea would be to use s, the sample standard
deviation, as an estimate to , the population standard
deviation.
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Introduction

• Even though we are primarily interested in ,
we are now forced to first estimate 
• We know that s changes from sample to
sample.
– So we are adding some variability into our
equations.

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Introduction

• When  is known,      n   is the standard
deviation of the sampling distribution of x-bar.
• When  is not known, it is estimated with
the sample standard deviation s
– Then we must estimate the standard deviation
of x-bar by:

– The standard error of the sample mean:
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Introduction
• Now, we also know that if x-bar is normal then
x
z     ~ N (0,1)
 n
s
• But if we use the standard error,    , does
n

x             ?
~ N (0,1)
s n

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Introduction

• Unfortunately it doesn’t. But it does follow a
distribution called the T-distribution.
x
t          ~ Tn-1
s/ n
• Where n-1 is the degrees of freedom
• From now on df = degrees of freedom
– Notice then that for each sample size there is a
different T-distribution.
• The degrees of freedom from this t statistic
come from the sample standard deviation s in
the denominator of the t statistic           8
History of the T distribution
• The T distributions were
discovered in 1908 by William
S. Gosset, a statistician
working for the Guinness
brewing company
• He published under the pen
name “Student” because
Guinness didn’t want
competitors to know that they
were gaining an industrial
statisticians
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Brilliant!
Properties of the T-distributions
• SIMILAR (but not the same) to a Normal dist.
– Symmetric
– Mean = 0
– Bell shaped
• **The spread of the tk distribution is a bit greater than
that of the standard Normal distribution
– This is due to the extra variability caused by substituting
the random variable s for the fixed parameter 
• As the degrees of freedom k increase, the tk density
curve approaches the N(0,1) curve
– This reflects the fact that s approaches  as the sample size
increases (Law of Large Numbers!!)
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Properties of the T-distributions
• The t distribution has more probability in the
tails and less in the center than does the
standard Normal distribution
• Table D in the back of the book gives critical
values for the t distributions
– For convenience, the table entries have also been
labeled by the confidence level C (in percent) required
for confidence intervals

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D Tea-table, anyone?

– Notice that it is not nearly as comprehensive
as the standard normal table
– This makes getting p-values a little more
difficult, but not much.

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The One-Sample t Confidence
Interval
• How does using s affect confidence intervals
for the mean  ?
• The one-sample t confidence interval is
similar in both reasoning and computational
detail to the z confidence interval of Chapter 6
– Note: “One-sample” does NOT mean that we
have a sample size of n=1

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The One-Sample t Confidence
Interval
• Suppose that an SRS of size n is drawn from a population
with an unknown mean  and standard deviation . A
level C confidence interval for  is:

– Where t* is the value for the Tn-1 density curve with area C
between -t* and t*.
– The margin of error is

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One-sample t confidence interval

The area between the critical
values –t* and t* under the
Tn-1 curve is C.                    Tn-1 Curve

Prob=(1-C)/2
Prob=(1-C)/2

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-t*        0    +t*
Example 1

• The following data are the amounts of vitamin
C, measured in milligrams per 100 grams of
blend, for a random sample of size 8 from a
production run:

26   31    23    22   11    22    14   31

• We want to find a 95% confidence interval for ,
the mean vitamin C content.

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n=8
x-bar = 22.5
s = 7.19
s    7.19
SE x            2.54
n     8
From Table D we find that for 95% CI t*7 = 2.365

 s                  7.19 
x t 
*
  22.5  2.365         (16.5, 28.5)
 n                  8 

We are 95% confident that the mean vitamin C content for this
production run is between 16.5 and 28.5 mg/100g.
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• In this example we have given the actual
interval (16.5, 28.5) as our answer

• Sometimes, we prefer to report the mean
and margin of error:
The mean vitamin C content is 22.5 mg/100g with
a margin of error of 6.0 mg/100g.

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Testing

• In tests of significance, as in confidence
intervals, we allow for unknown  by using s
and replacing z by t
• Let n be the sample size
x
– If σ is known and n is large then z            ~ N (0, 1)

n
– If σ is NOT known then
x
t     ~ Tn 1
s
n                     19
The one-sample t test
• Suppose that a SRS of size n is drawn from a
population having unknown mean 

• To test the hypothesis H0 :  = 0 based on a
SRS of size n, compute the one-sample t
statistic:
x  0
t
s n
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The one-sample t test
• The P-value for a test of H0 against

– Ha :  > 0 is P(Tn-1  t)

– Ha :  < 0 is P(Tn-1  t)

– Ha :   0 is 2P(Tn-1  |t|)

• These P-values are exact if the population
distribution (X) is Normal and are approximately
correct for large n in other cases              21
…Example 1

• Recall that n = 8, x-bar = 22.50, and s = 7.19

• Suppose we want to test whether the mean vitamin
C content in the final product is 40

• Hypotheses:
H0: µ = 40
Ha: µ ≠ 40
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…Example 1
• Test statistic:
x  0   22 .5  40
t                     6.88
s n       7.2 8
• P-value:

2P T7  6.88   2P T7  5.408 
2P T7  5.408   2(.0005)  0.001  P  value  0.001

• Decision and Conclusion:
– We reject H0
– There is significant evidence to conclude that the
mean vitamin C content for this run is not equal to 40.   23
Two-sided alternative and CI’s

• We can use (1-)% Confidence intervals
to test a two-sided alternative alternative
hypothesis at the  significance level
– If the confidence contains the null mean (the
mean that is assumed to be true) then this is
a plausible value and fail to reject H0.

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