Transcript 8.3x

Section 8.3
Estimating Population Means
( Unknown)
And some valuable added stuff
by D.R.S., University of Cordele
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Should I use z or t for my confidence interval?
• We have two bell-shaped distributions, z and t.
• Which one to use?
Do I have a
“big” sample?
𝑛 ≥ 30
no
yes
Is the
population
approximately
normally
distributed?
no
I need more
advanced
techniques that
are beyond the
scope of this
course.
yes
yes
Use
z
Do I know the value of no
σ, the population’s
standard deviation?
Use
t
The formula for E in a t problem is
very much like the formula for E in a z problem
They have the same arithmetic structure.
𝐸, the
margin
of error:
Difference: using a t
critical value instead
of a z critical value.
Difference: using sample
standard deviation s
instead of population
standard deviation σ.
Same:
square root of the
sample size, 𝑛
Estimating Population Means ( Unknown)
Confidence Interval for a Population Mean
The confidence interval for a population mean is given
by
x E   x E
or
x  E, x  E
Where x is the sample mean, which is the point
estimate for the population mean, and E is the margin
of error.
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The train crossing at Highway 92 was observed one day.
Thirteen trains passed and the length of time the
roadway was blocked was recorded for each. The mean
was 282 sec (4 min 42 sec) and the standard deviation
was 100 sec (1 min 40 sec). The distribution of all
trains’ lengths is thought to be normal. Construct the
90% confidence interval of the time it takes for a train
to pass. Is this a z problem or a t problem? ______
Why is it legitimate to use these formulas?
Because even though ___________, the
population has ___________ ____________.
Make note of variables and values:
_____ = 13
_____ = 282
_____ = 100
c = _____
α = _____
α / 2 = _____
t α / 2 = _____
To replay the entire example,
go to http://2205.drscompany.com,
click on Examples,
click on Chapter 8,
click on Section 8.3,
click on “Confidence interval with t”
Compute the Margin of Error:
Compute the Confidence Interval:
x E   x E
or
x  E, x  E
Verify by doing the problem with the TI-84 Tinterval.
Example 8.16: Constructing a Confidence
Interval for a Population Mean ( Unknown)
A marketing company wants to know the mean price of
new vehicles sold in an up-and-coming area of town.
Marketing strategists collected data over the past two years
from all of the dealerships in the new area of town. From
previous studies about new car sales, they believe that the
population distribution looks like the following graph.
Valid sample? _______
Normal distribution? ________
Use z or t? _______
Need a big sample, 30 or more? _____
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Example 8.16: Constructing a Confidence Interval for
a Population Mean ( Unknown) (cont.)
The simple random sample of 756 cars has a mean of
$27,400 with a standard deviation of $1300. Construct
a 95% confidence interval for the mean price of new
cars sold in this area.
Make some notes as you read:
756 = _____
$27400 = _______
$1300 = ______
95% = ______ so _____ = ______ and ____ / 2 = ______
Should we use z or use t or do we need a technique that’s
not part of this course? __________________________
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Example 8-16 – Car Prices, continued
Compute the margin of error, E. You’ll need d.f. = _____.
Determine the confidence interval, 𝑥 − 𝐸, 𝑥 + 𝐸 :
State a conclusion in plain English: “We are _____%
confident that __________________________________
is between $__________ and $__________
Example 8-16 – Car Prices, continued
Do the same problem again,
but this time use TI-84
STAT, TESTS, 8:TInterval
What does the TI-84 give
for the confidence interval?
It does not tell you the margin of error, E, directly.
But could you figure out E from the information
shown?
Further words about ZInterval and TInterval
• If you’re asked for a confidence interval,
• Use ZInterval for a normal distrib. situation.
• Use TInterval for a t-distribution situation.
• If the problem asks for a critical value of z or t, too,
• Then you have to use invNorm( or invT( or a
printed table to answer that question.
• Make the right choice between
• Stats, if you’re given the mean, etc.
• Data, if you’re given a list of raw data
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Systems/Quant Systems, Inc.
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Example 8.18: Constructing a Confidence Interval for a
Population Mean ( Unknown) from Original Data
Given the following sample data from a study on the
average amount of water used per day by members of
a household while brushing their teeth, calculate the
99% confidence interval for the population mean using
a TI-83/84 Plus calculator. Assume that the sample
used in the study was a simple random sample.
Should we use z or t or some other advanced
technique? And why?
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Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
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Example 8.18: Constructing a Confidence Interval for a
Population Mean ( Unknown) from Original Data (cont.)
*
Household Water Used for Brushing Teeth (in Gallons per Day)
0.485
0.428
0.39
0.308
0.231
0.587
0.516
0.465
0.370
0.282
0.412
0.367
0.336
0.269
0.198
0.942
0.943
0.940
0.941
0.946
0.868
0.898
0.889
0.910
0.927
0.925
0.950
0.959
0.948
0.956
0.805
0.810
0.839
0.860
0.861
0.515
0.463
0.420
0.326
0.243
HAWKES LEARNING SYSTEMS
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Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 8.18: Constructing a Confidence Interval for a
Population Mean ( Unknown) from Original Data (cont.)
To begin with, since we are given the raw data and not
the sample statistics, we need to enter the data in the
calculator list, like in L1. Then use TInterval, but this
time highlight the Data option, not the Stats option!
You’ll see some differences in the prompts.
Confidence interval result is
( _________, ___________ )
Conclusion: _____ % confident
that ___________________
_______________________
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Stuff you need to know about
the Practice and Certify problems
• They come in pairs.
• First part asks you for “the critical value”.
• Second part asks you for the confidence
interval.
• For the critical value, the easiest thing to do is to use
your printed tables.
• invT can be used but the tables area easier.
Stuff you need to know about
the Practice and Certify problems
• For the Confidence Interval, TInterval with the
calculator is the best way. The Tutor probably does
it the long way. There are some multipart problems
that give you the raw data. Put the data into a TI84 list.
• Use 1-Var Stats to answer any preliminary
questions about mean and standard deviation.
• Use T-Interval with “Data”, not “Stats”, to get
the confidence interval.