Download Confidence Interval Estimation for a Population Mean

Confidence Interval
Estimation for a
Population Mean
Lecture 33
Section 10.3
Tue, Nov 14, 2006
Confidence Intervals
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To estimate , we will use confidence intervals,
as we did when estimating p.
The basic form, as well as the theory, is the
same:
(pt. est.)  (approp. no. of st. devs.)
Confidence Intervals
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What is the point estimate for ?
What is the standard deviation for this
estimator?
How do we determine the appropriate number
of standard deviations?
Confidence Intervals
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Ifx has a normal distribution, then the
confidence interval is
or
x  z  / n
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x  z  s / n
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If (x – )/(s/n) has a t distribution, then the
confidence interval is

x t s/ n

When to Use Z
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If
The population is normal (or nearly normal) and  is
known, or
 The population is not normal, but the sample size is
at least 30,
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Then use Z.
When to Use t
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If
The population is normal (or nearly normal), and
  is not known, and
 The sample size is less than 30,
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Then use t.
Example
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Example 10.4, p. 641: The Kellogg Corporation
controls approximately a 43% share of the ready-to-eat
cereal market worldwide. A popular cereal is Corn
Flakes. Suppose the weights of full boxes of a certain
kind of cereal are normally distributed with a
population standard deviation of 0.29 ounces. A
random sample of 25 boxes produced a mean weight
of 9.82 ounces.
Construct a 95% confidence interval for the true mean
weight of such boxes.
Example
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Use Z. Why?
n = 25.
x = 9.82.
Assume that  = 0.29.
Level of confidence = 95%, so z = 1.96.
Example
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The confidence interval is
9.82  (1.96)(0.29/25)
= 9.82  0.114
= (9.706, 9.934).
TI-83 – Confidence Intervals
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When the standard normal distribution applies,
do the following.
Press STAT.
Select TESTS.
Select ZInterval.
A window appears requesting information.
TI-83 – Confidence Intervals
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Select Data or Stats.
Assume we selected Stats.
Enter .
Enterx.
Enter n.
Enter the level of confidence.
Select Calculate and press ENTER.
TI-83 – Confidence Intervals

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A window appears containing
The title “ZInterval”.
The confidence interval in interval notation.
The sample mean.
The sample size.
Example
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Example 10.5, p. 643: Unoccupied seats on flights
cause airlines to lose revenue. Suppose that a large
airline wants to estimate its average number of
unoccupied seats per flight from Detroit to
Minneapolis over the past month. To accomplish this,
the records of 61 such flights were randomly selected,
and the number of unoccupied seats was recorded for
each of the sampled flights. The sample mean is 12.6
and sample standard deviation is 4.4 seats.
Construct a 99% confidence interval for the mean
number of unoccupied seats.
Example
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Should we use Z or t? Why?
n = 61.
x = 12.6.
s = 4.4.
Level of confidence = 99%. Find t.
Example
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Consider again the t table (Table IV).
The degrees of freedom include every value up
to 30, then jump to 40, 60, 120.
If the actual degrees of freedom are
Between 30 and 40, use 30.
 Between 40 and 60, use 40.
 Between 60 and 120, use 60.
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If they are beyond 120, use z.
Example
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The confidence interval is
12.6  (2.660)(4.4/61)
= 12.6  1.499
= (11.101, 14.099).
TI-83 – Confidence Intervals
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To use t, do the following.
Press STAT.
Select TESTS.
Select TInterval.
A window appears requesting information.
TI-83 – Confidence Intervals
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





Select Data or Stats.
Assume we selected Stats.
Enterx.
Enter s.
Enter n.
Enter the level of confidence.
Select Calculate and press ENTER.
TI-83 – Confidence Intervals





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A window appears containing
The title “TInterval”.
The confidence interval in interval notation.
The sample mean.
The sample standard deviation.
The sample size.