Download Previously, when making inferences about the population mean

Chapter 18
Inference about a Population Mean
Conditions for inference
Previously, when making inferences about the population mean, , we were
assuming the following simple conditions:
(1)
Our data (observations) are a simple random sample (SRS) of size n
from the population of interest.
(2)
The variable we measure has an exactly normal distribution with
parameters  and .
(3)
Population standard deviation  is known.
Then we were constructing confidence interval for the population mean  based
on _________ distribution (one-sample z statistic):
This holds approximately for large samples even if the assumption (2) is not
satisfied. Why?
Issue: In a more realistic setting, assumption (3) is not satisfied, i.e., the
standard deviation  is unknown.
So what can we do to handle real-life problems?
We replace the population standard deviation,  by its estimate:
When σ is known, the standard deviation of the sample mean x is
When σ is unknown, we then estimate the standard deviation of x by
(This quantity is called the _____________ of the sample mean x .)
We get the one-sample t statistic:
When making inferences about the population mean  with  unknown we use
the one-sample t statistic (Note that we still need the assumptions 1 and 2).
But one-sample t statistic doesn’t have normal distribution, it has
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The t-distributions
We specify a particular t-distribution by giving its degrees of freedom (d.f.).
How does t-distribution compare with standard normal distribution?
Similarities:
Difference:
As the d.f. k increases, the tk distribution approaches the Normal(0,1) distribution.
Notation: tk represents the t-distribution with k d.f.
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Confidence Intervals for a Population Mean (when standard deviation σ
is unknown)
Confidence interval for  when  is unknown (t -CI)
A level C confidence interval for  is given by
where t* is the upper (1-C)/2 critical value for the tn-1 distribution, i.e.,
Ex: What critical value t* from Table C would you use to make a CI for the
population mean in each of the following situations?
a) A 95% CI based on n = 10 observations.
b) A 90% CI based on n = 26 observations.
c) An 80% CI from a sample of size 7.
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Ex: Suppose the JC-Penney wishes to know the average income of the
households in the Dallas area before they decide to open another store here. A
random sample of 21 households is taken and the income of these sampled
households turns out to average $45,000 with a standard deviation of $15,000.
Give a 90% confidence interval for the unknown average income of the
households in Dallas area.
Steps for doing test of significance about  when  is unknown (t-test)
1. State the null and the alternative hypotheses.
2. Calculate the one-sample t statistic (assuming H0 to be true):
3. Calculate P-value and use it to draw conclusion.
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P-values for the t-test
Ha
P-value
μ > μ0
Pr(T ≥ t)
μ < μ0
Pr(T ≤ t)
μ  μ0
Pr(T≥|t|) + Pr(T≤|t|)
= 2 Pr(T≥|t|)
Area under curve
Exact P-values cannot be obtained using Table C. Locate the row corresponding
to the n-1 degrees of freedom. Slide across until you find the critical values that
your test statistic falls between. Use the corresponding upper tail probabilities at
the top of the table to calculate an interval in which the P-value falls.
Ex: Suppose testing H0:  = 0 vs Ha:  > 0 yields a one-sample t statistic of 1.82
from a sample of size 15.
a) What are the degrees of freedom for this statistic?
b) Give two critical values t* from Table C that bracket t. What are the right-tail
probabilities for these two entries?
c) Between what two values does the P-value fall?
d) Is t = 1.82 statistically significant at the 5% level? At 1%?
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Ex: Bottles of Boca Cola are supposed to contain 300 ml of cola. A consumer
group is suspicious that the Boca Cola bottles contain less cola than what is
advertised. In order to check their suspicion, they measure the contents of 9
randomly selected bottles. They find an average cola content of 299.6 ml and a
standard deviation of 0.4 ml. Carry out the appropriate test of significance.
a) State the null and the alternative hypotheses.
b) Compute the one-sample t statistic.
a) What are the degrees of freedom for the above statistic?
b) Between what two probabilities p from the Table C does the P-value of the test
fall. What do you conclude at 1% level?
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Matched Pairs t Procedures
As we mentioned in Chapter 9, comparative studies are more convincing than
single-sample investigations. For that reason, one sample- inference is less
common than comparative inference.
In a matched pairs design, subjects are matched in pairs and each treatment is
given to one subject in each pair. The experimenter can toss a coin to assign two
treatments to the two subjects in each pair.
Example 1. Suppose a college placement center wants to estimate µ, the
difference in mean, starting salaries for men and women graduates who seek jobs
through the center. If it independently samples men and women, the starting
salaries may vary because of their different college majors and differences in
grade point averages. To eliminate these sources of variability, the placement
center could match male and female job-seekers according to their majors and
GPAs. Then the differences between the starting salaries of each pair in the
sample could be used to make an inference about µ.
Example 2. Suppose you wish to estimate the difference in mean absorption rate
into the bloodstream for two drugs that relieve pain. If you independently sample
people, the absorption rates might vary because of age, weight, sex, etc. It may
be possible to obtain two measurements on the same person. First, we administer
one of the two drugs and record the time until absorption. After a sufficient
amount of time, the other drug is administered and a second measurement on
absorption time is obtained. The differences between the measurements for each
person in the sample could then be used to estimate µ.
Another situation calling for matched pairs is before-and-after observations on the
same subjects.
Example 3. Suppose you wish to estimate the difference in mean blood pressure
before and after taking a drug. We will obtain the first measurement before a
patient is taking the drug and second measurement after a sufficient amount of
time that the patient was taking the drug. The differences between the
measurements for each person in the sample could then be used to estimate µ.
If the samples are matched pairs, find the difference between the
responses within each pair, then apply one-sample t procedures to
those differences of observed responses.
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Example. An experiment is conducted to compare the starting salaries of male and
female college graduates who find jobs. Pairs are formed by choosing a male and
a female with the same major and similar GPA. Suppose a random sample of 10
pairs is formed in this manner and the starting annual salary of each person is
recorded. Let µ1 be the mean starting salary for males and let µ2 be the mean
starting salary for females.
Pair
1
2
3
4
5
6
7
8
9
10
Male (in $)
29300
41500
40400
38500
43500
37800
69500
41200
38400
59200
Female (in $)
28800
41600
39800
38500
42600
38000
69200
40100
38200
58500
Difference (male – female)
500
- 100
600
0
900
- 200
300
1100
200
700
(a) Compute a 95% confidence interval for the mean difference µ = µ1-µ2.
The sample average of the paired difference
x
and the sample standard deviation of the paired difference
s
The 95% paired difference CI for  = 1-2 is
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Robustness of t procedures
A confidence interval is called robust if the confidence level does not change very
much when the conditions for use of the procedure are violated.
The t confidence interval is exact when the distribution of the population is exactly
_________. However, no real data are exactly ________.
The usefulness of the t procedures in practice therefore depends on
Here are some practical guidelines for inference on population means:
***Always make a plot to check for skewness and outliers before using the t
procedures for small samples. ***
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Using the t procedures
 Except in the case of small samples, the condition that the data are an SRS
from the population of interest is more important than the condition that the
population distribution is normal.
 Sample size less than 15: Use t procedures if the data appear close to
normal (roughly symmetric, single peak, no outliers). If the data are clearly
skewed or if outliers are presented, do not use t procedures.
 Sample size at least 15: The t procedures can be used except in the
presence of outliers or strong skewness.
 Large samples: The t procedures can be used even for clearly skewed
distributions when the sample size is large, say n ≥ 40.
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