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STAT 305: Chapter 10 – Inference for a Single Population Mean (Y
Spring 2014
10.1 – Confidence Interval for a Single Population Mean (  )
We have already looked at confidence intervals as a method of making decisions
about the population mean . The CI details are summarized below.
(100 -  )% Confidence Interval for  (e.g.   .05  95% confidence)
The basic form of a confidence interval is as follows:
(estimate) + (table value) * SE(estimate)
For a single population mean a 100(1-  )% CI for  is:
y  t SE ( y )
where SE ( y )  s
Confidence Level
95 % (   .05)
90 % (   .10 )
99 % (   .01 )
n
and t = t-distribution quantile with df  n  1 .
To find the t-statistic
value ( t ) use the twotail at the top of the
columns in the table.
Before we look at hypothesis testing for a single population mean we will
examine the five basic steps in a hypothesis test and introduce some important
terminology and concepts.
10.2 - Steps in a Hypothesis Test (in general)
1.
2.
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3.
4.
5.
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10.3 - Hypothesis Testing for a Population Mean (  )
General Form of Hypotheses for a Population Mean:
Null Hypothesis ( H o )
Alternative Hypothesis ( H a )
p-value area
H o :   o
H a :   o
Upper-tail
H o :   o
H a :   o
Lower-tail
H o :   o
H a :   o
Two-tailed
(can perform test using CI for  )
 o  hypothesized value for the mean assuming the null hypothesis is true.
Examples of Formulating Hypotheses:
1) Suppose in the past the north end of a particular lake had a Secchi depth of 8.0 meters.
Due to recent development around the lake, researchers believe that there is decreased
water clarity in that section of the lake. Set up the appropriate hypotheses for this
situation.
2) In the community of Morgan Hill, CA there is concern about the perchlorate level
found in well water. EPA guidelines suggest that a water supply should have a mean
perchlorate level below 4 ppb (parts per billion). Environmental scientists wish to
determine if mean perchlorate level in the Morgan Hill water supply is greater than the
safe limit. Set up the appropriate hypotheses for this situation.
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Test Statistic – in general
In general the basic form of most test statistics is given by:
(estimate)  (hypothesized value)
Test Statistic =
(think “z-score”)
SE (estimate)
which measures the discrepancy between the estimate from our sample and the
hypothesized value under the null hypothesis relative to the size of the standard error.
Intuitively, if our sample-based estimate is “far away” from the hypothesized value
assuming the null hypothesis is true, we will reject the null hypothesis in favor of the
alternative or research hypothesis. Extreme test statistic values occur when our estimate
is a large number of standard errors away from the hypothesized value under the null.
The p-value is the probability, that by chance variation alone, we would get a test statistic
as extreme or more extreme than the one observed assuming the null hypothesis is true.
If this probability is “small” then we have evidence against the null hypothesis, in other
words we have evidence to support our research hypothesis.
Test Statistic for Testing a Single Population Mean (  ) ~ (t-test)
y  o
y  o
t
or t 
~ t-distribution with df = n – 1.
s
SE ( y )
n
Assumptions:
When making inferences about a single population mean we assume the following:
1. The sample constitutes a random sample from the population of interest.
2. The population distribution is normal. This assumption can be relaxed when
our sample size in sufficiently “large”. How large the sample size needs to be is
dependent upon how “non-normal” the population distribution is.
Type I and Type II Errors ( &  )
Truth
Decision
H o true
H a true
Reject H o
Fail to
Reject H o
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Type I and II Errors Example:
Testing Wells for a Perchlorate in Morgan Hill & Gilroy, CA.
EPA guidelines suggest that drinking water should not have a perchlorate level exceeding
4 ppb (parts per billion). Perchlorate contamination in California water (ground, surface,
and well) is becoming a widespread problem. The Olin Corp., a manufacturer of road
flares in the Morgan Hill area from 1955 to 1996 was is the source of the perchlorate
contamination in the this area.
Suppose you are resident of the Morgan Hill area which alternative do you want well
testers to use and why? (Which has the more serious type I error, A or B?)
A
B
H o :   4 ppb
H o :   4 ppb
or
H a :   4 ppb
H a :   4 ppb
10.4 - Examples: Testing a Single Population Mean ()
Example 10.1: Secchi Disc Readings in Seneca Lake (New York)
In 1997 the mean Secchi depth recorded at the north end of Seneca Lake was 8.0 meters.
In 1999 a depth study was conducted on the north end of the lake on similar dates as in
the 1997 study yielding the data below:
9.0
9.0
10.5 8.0
9.2
7.0
8.3
6.0
7.1
4.7
7.8
6.0
7.9
6.5
8.3
7.4
Is there evidence to suggest the Secchi depth has decreased on the north end of the lake?
From the JMP output we see the following:
 Normality appears to be satisfied here.
 Notice the CI for the mean Secchi depth is (6.89 m, 8.44 m).
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Hypothesis Test:
1) State hypotheses
2) Choose 

Test statistic
3) Compute test statistic
4) Find p-value (use t-Probability Calculator.JMP)
5) Make decision and interpret
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Performing the t-Test in JMP:
To perform a t-test in JMP, select Test Mean from the Secchi Depth pull-down
menu and enter value for mean under the null hypothesis, 8.0 in this example.
This will give us the value of the t-test statistic as well as p-values from all three
possible alternative hypotheses.
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Example 10.2: Testing Wells for a Perchlorate in Morgan Hill & Gilroy, CA
Hypothesis Test:
1) State hypotheses
2) Choose 
Test statistic
3) Compute test statistic
4) Find p-value (use t-Probability Calculator.JMP)
5) Make decision and interpret
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Performing the t-Test in JMP:
To perform the t-test in JMP, select Test Mean from the Perchlorate pull-down
menu and enter value for mean under the null hypothesis, 4.0 in this example.
Conclusion:
Q: Should we be using the mean as a measure of the typical perchlorate
concentration found in these wells? Why or why not?
Let’s consider the log base 10 of the perchlorate levels instead.
Hypothesis Test in Log Scale:
Test results from JMP
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Confidence Interval in the Log Base 10 Scale
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