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It is important that the "standard deviation" used must be appropriate for the parameter we are estimating, so in this section we need to use the standard deviation that applies to sample means, which is σ n . The fraction σ n , is commonly called the "standard error of the mean" in order to distinguish clearly the standard deviation for a mean from the population standard deviation σ .

    In summary, as a result of the central limit theorem:

  • X ¯ is normally distributed, that is, X ¯ ~ N ( μ X , σ n ) .
  • When the population standard deviation σ is known, we use a normal distribution to calculate the error bound.

Calculating the confidence interval

To construct a confidence interval estimate for an unknown population mean, we need data from a random sample. The steps to construct and interpret the confidence interval are:

  • Calculate the sample mean x ¯ from the sample data. Remember, in this section we already know the population standard deviation σ .
  • Find the z -score that corresponds to the confidence level.
  • Calculate the error bound EBM .
  • Construct the confidence interval.
  • Write a sentence that interprets the estimate in the context of the situation in the problem. (Explain what the confidence interval means, in the words of the problem.)

We will first examine each step in more detail, and then illustrate the process with some examples.

Finding the z -score for the stated confidence level

When we know the population standard deviation σ , we use a standard normal distribution to calculate the error bound EBM and construct the confidence interval. We need to find the value of z that puts an area equal to the confidence level (in decimal form) in the middle of the standard normal distribution Z ~ N (0, 1).

The confidence level, CL , is the area in the middle of the standard normal distribution. CL = 1 – α , so α is the area that is split equally between the two tails. Each of the tails contains an area equal to α 2 .

The z-score that has an area to the right of α 2 is denoted by z α 2 .

For example, when CL = 0.95, α = 0.05 and α 2 = 0.025; we write z α 2 = z 0.025 .

The area to the right of z 0.025 is 0.025 and the area to the left of z 0.025 is 1 – 0.025 = 0.975.

z α 2  =  z 0. 025  = 1 .96 , using a calculator, computer or a standard normal probability table.

Note

Remember to use the area to the LEFT of z α 2 ; in this chapter the last two inputs in the invNorm command are 0, 1, because you are using a standard normal distribution Z ~ N (0, 1).

Calculating the error bound ( EBM )

The error bound formula for an unknown population mean μ when the population standard deviation σ is known is

  • EBM = ( z α 2 ) ( σ n )

Constructing the confidence interval

  • The confidence interval estimate has the format ( x ¯ E B M , x ¯ + E B M ) .

The graph gives a picture of the entire situation.

CL + α 2 + α 2 = CL + α = 1.

This is a normal distribution curve. The peak of the curve coincides with the point x-bar on the horizontal axis. The points x-bar - EBM and x-bar + EBM are labeled on the axis. Vertical lines are drawn from these points to the curve, and the region between the lines is shaded. The shaded region has area equal to 1 - a and represents the confidence level. Each unshaded tail has area a/2.

Writing the interpretation

The interpretation should clearly state the confidence level ( CL ), explain what population parameter is being estimated (here, a population mean ), and state the confidence interval (both endpoints). "We estimate with ___% confidence that the true population mean (include the context of the problem) is between ___ and ___ (include appropriate units)."

Questions & Answers

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Damian Reply
absolutely yes
Daniel
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Maciej
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Abigail
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
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Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
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abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
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I'm interested in nanotube
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what is nanomaterials​ and their applications of sensors.
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what is system testing?
AMJAD
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Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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anybody can imagine what will be happen after 100 years from now in nano tech world
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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Prasenjit
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Damian
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Damian
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Azam
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Uday
I'm interested in Nanotube
Uday
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Prasenjit
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Source:  OpenStax, Statistics i - math1020 - red river college - version 2015 revision a - draft 2015-10-24. OpenStax CNX. Oct 24, 2015 Download for free at http://legacy.cnx.org/content/col11891/1.8
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