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Confidence Intervals
A Refresher Before Constructing Confidence Intervals
1. Statistics describe samples; Parameters describe populations
2. Each parameter has a corresponding statistic that serves as a point estimate of the associated
parameter (see Table 1 below).
3. Point estimates are just what the name implies: it is a single point associated with a sample that
serves as the estimate of the corresponding parameter. By itself it is a weak estimate of the
parameter because it doesn‟t take into account the inherent uncertainty of how close it is to the
real value of the parameter.
4. The accuracy of the point estimate as a predictor of the corresponding parameter can be
significantly strengthened by the addition of a margin of error, which is often represented as E. By
adding E to the point estimate and then subtracting E from the point estimate, an interval is
formed with an upper and lower bound. This interval is said to contain the associated parameter
with a certain degree of confidence, and is much more useful than the point estimate alone.
5. The margin of error E is a function of sample size, sample standard deviation, and a critical value
zc or tc that is associated with a level of confidence that the interval does in fact contain the
parameter in question.
6. Examine the following equation for margin of error:
√
This is the formula one would
use with a large sample. As illustrated in Table 2, when the required confidence level increases,
so does the value of zc, which in turn increase the value of E. In other words, a demand for a
higher confidence level will necessarily lead to a higher margin of error, which results in wider
confidence intervals. Also look at n, the sample size. Since it is in the denominator, as n
increases, the margin of error E decreases. This is intuitive; the bigger the sample size pulled out
of the population, the better the sample represents population parameters in question.
Unfortunately, sample size is often costly to increase, so a balance is often struck by the
statistician between the size of the sample and the value of the confidence level (which drives Z c).
Point Estimate of the Parameter
̅ (X-BAR)
MEAN OF THE SAMPLE
̂ (P-HAT)
Parameter
µ
MEAN OF THE POPULATION
P
PROPORTION OF THE SAMPLE
PROPORTION OF THE POPULATION
S
σ
STANDARD DEVIATION OF THE SAMPLE
STANDARD DEVIATION OF THE POPULATION
S
2
VARIANCE OF THE SAMPLE
σ2
VARIANCE OF THE POPULATION
Table 1. Relationship between Point Estimate (statistic derived from a sample) and Parameter
Confidence Intervals
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Case 1: Confidence Intervals for the Mean 𝛍: Large Sample (N>30)
Five Step Manual Process
1. Find the sample statistics
̅ and n
̅
2. Specify σ, if known. Otherwise use the sample
standard deviation s as an estimator of σ since
n>30.
3. Find the critical value Zc that corresponds to the
given level of confidence.
̅
√
See Table 2 below, or use the Standard
Normal Table.
4. Find the Margin of Error E
√
Left endpoint: ̅
Right endpoint: ̅
̅
Interval:
5. Find the left and right endpoints, and form the
confidence interval.
Confidence Level*
Alpha (∝)
Alpha (∝/2)
Zα/2or Zc
.90 (90%)
.95 (95%)
.98 (98%)
.10
.05
.02
.05
.025
.01
1.645
1.96
2.33
.99 (99%)
.01
.005
2.575
̅
Table 2. Some common Confidence Levels and corresponding Zα/2
Example 1: A random sample of 32 gas grills has a mean price of 630.90 and a standard
deviation of $56.70. Construct confidence intervals for the mean µ using 90% and 95% levels of
confidence.
1.
2.
3.
4.
Sample statistics:
Sample standard deviation
Zcritical: See Table 2
Find E
̅
and n=32
s= 56.70
Zc=1.645 (90%); 1.96 (95%)
√
√
5. Endpoints and Interval
√
√
= 16.49 (90%)
= 19.65 (95%)
Left Endpoint: 630.90 - 16.49=614.41
Right Endpoint: 630.90 + 16.49=647.39
90% interval: 614.41 < µ < 647.39
Left Endpoint: 630.90 - 19.65=611.25
Right Endpoint: 630.90 +19.65=650.55
95% interval: 611.25 < µ < 650.55
Confidence Intervals
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Example 2: Re-do Example 1 using either the TI-83 or TI-84 calculator
1. Go to STATS, TESTS, ZInterval
Enter data for 90% confidence level
2. Press Calculate
1. Go to STATS, TESTS, ZInterval
Enter data for 95% confidence level
2. Press Calculate
As can be seen above, use of the ZInterval test (because n>30) on the TI-83/84 yields the same
answer as the manual method, in a fraction of the time.
Case 2: Confidence Intervals for the Mean 𝛍: Small Sample (N<30)
Four Step Manual Process
1. Find the sample statistics
̅ ,s, and n
2. Identify the degrees of freedom, the level of
confidence, and the critical value tc
3. Find the Margin of Error E
4. Find the left and right endpoints, and form the
confidence interval.
̅
√
̅
d.f. = n-1
√
Left endpoint: ̅
Right endpoint: ̅
̅
Interval:
̅
Confidence Intervals
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Example 3: A random sample of 5 microwave ovens has a mean repair price of $75.00 and a
standard deviation of $12.50. Construct confidence intervals for the mean µ using a 95% level of
confidence.
1. Find the sample statistics
̅
̅ ,s, and n
2. Identify the degrees of freedom, the level of
confidence, and the critical value tc
d.f. = n-1 = 4
√
√
3. Find the Margin of Error E
4. Find the left and right endpoints, and form
the confidence interval.
From t-distribution table:
tc for df=4; 2-tail confidence level .95, is 2.776
Left endpoint: ̅
Right endpoint: ̅
Interval: 59.48
Example 4: Re-do Example 3 using either the TI-83 or TI-84 calculator
1. Go to STATS, TESTS, TInterval
Enter data for 95% confidence level
2. Press Calculate
As can be seen above, use of the TInterval test (because n<30) on the TI-83/84 yields the same
answer as the manual method, in a fraction of the time.
Confidence Intervals
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Case 3: Confidence Intervals for Population Proportions
Six Step Manual Process
1. Find the sample statistics n and x
2. Find the point estimate ̂
3. Verify that the sampling distribution of ̂ can
be approximated by the normal distribution
4. Find the critical value zc that corresponds to
the given level of confidence.
̂
̂
Use Table 2 on page 2 of this Handout
or the Standard Normal Table
5. Find the margin of error E
6. Find the left and right endpoints, and form
the confidence interval.
̂
√
q
Left endpoint: ̂
Right endpoint: ̂
̂
Interval:
̂
Example 5: 2,563 adults were asked if they believed that the activities of humans are contributing
to an increase in global temperatures. 1,666 replied „Yes‟. Construct a 99% confidence interval
for the proportion of adults who believe that the activities of humans are contributing to an
increase in global temperatures.
7. Find the sample statistics and n and x
n= 2,563; x= 1666
8. Find the point estimate ̂
̂
9. Verify that the sampling distribution of ̂ can
be approximated by the normal distribution
10. Find the critical value zc that corresponds to
the given level of confidence.
11. Find the margin of error E
12. Find the left and right endpoints, and form
the confidence interval.
̂
̂
2.575
√
̂̂
√
Left endpoint: ̂
Right endpoint: ̂
Interval:
0.626
Confidence Intervals
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Example 6 Re-do Example 5 using either the TI-83 or TI-84 calculator
1. Go to STATS, TESTS, 1-PropZint
Enter data for 99% confidence level
2. Press Calculate
As can be seen above, use of the 1-PropZint test on the TI-83/84 yields the same answer as the
manual method, in a fraction of the time.
Case 4: Confidence Intervals for Variance and Standard Deviation
Six Step Manual Process
1. Verify that the population has a normal
distribution
2. Identify the sample statistic n and the
degrees of freedom.
3. Find the point estimate s
̅
2
4. Find the critical values
that
correspond to the given level of confidence.
Use Chi-Square Distribution Table
Left endpoint:
5. Find the left and right endpoints, and form
the confidence interval for the population
variance.
Right endpoint:
Interval:
6. Find the confidence interval for the
population standard deviation by taking the
square root of each endpoint.*
Interval:
√
√
*Reminder: Standard Deviation = √
Confidence Intervals
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Example 7: A lawn mower manufacturer is trying to determine the standard deviation of the life of one of
its lawn mower models. To do this, it randomly selects 12 lawn mowers that were sold several years ago
and finds that the sample standard deviation is 3.25 years. Construct a 99% confidence interval for the
standard deviation. Assume a normal distribution.
Six Step Manual Process
1. Verify that the population has a normal
distribution
Given
2. Identify the sample statistic n and the
degrees of freedom.
3. Find the point estimate s
2
2
(3.25) =10.5625
4. Find the critical values
that
correspond to the given level of confidence.
26.757, 2.603
Left endpoint:
5. Find the left and right endpoints, and form
the confidence interval for the population
variance.
Right endpoint:
Interval:
4.34 <
Interval:
6. Find the confidence interval for the
population standard deviation by taking the
square root of each endpoint.*
*Reminder: Standard Deviation = √
Interval:
√
< 44.64
√
Interval:
Confidence Intervals
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Document created by South Campus Library Learning Commons 03/26/10. Permission to copy and use is
granted for educational use provided this copyright label is displayed.
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Example 8 Re-do Example 7 using either the TI-83 or TI-84 calculator
Note:
In its current configuration, the TI-83/84 does not have the capability to construct confidence
intervals for variances and standard deviations.
However, a program designed specifically for the TI-83/84 by Michael Lloyd of Henderson
State University does have this capability, and is available as a free download from
www.aw.com/triola . The two programs that must be downloaded from this site are “S2INT”
and “ZZINEWINT”.
After they are downloaded, extract the contents to a folder on your computer, and then
upload both extracted files to the TI-83/84. The upload process will require a TI Connect
cable, and the free TI Connect software available at the Texas Instrument site.
The following is the solution to Example 7 using the aforementioned software.
(Assuming “S2INT” and “ZZINEWINT” have been
uploaded to the TI-83/84 and tested)
1. Select PRGM
2. Select S2INT
(see top screen to the right)
3. Hit ENTER twice
4. Complete data entry
nd
(see 2 screen to the right)
5. Hit ENTER; Read Variance Confidence Interval
rd
(see 3 screen to the right)
7. Hit ENTER; Read Std Dev Confidence Interval
(see bottom screen to the right)
Use of this program when installed on the TI-83/84 can make quick work of the task of
constructing confidence intervals for variances and standard deviations. As can be seen in the
bottom two screen shots, the user will have to use the arrow keys to move to the right to pick up
the rest of the solution.
Confidence Intervals
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Document created by South Campus Library Learning Commons 03/26/10. Permission to copy and use is
granted for educational use provided this copyright label is displayed.