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This module provides a practice of Hypothesis Testing of Single Mean and Single Proportion as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

Student learning outcomes

  • The student will conduct a hypothesis test of a single mean with known population standard deviation.


Suppose that a recent article stated that the mean time spent in jail by a first–time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first–time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was 3 years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. Conduct a hypothesis test to determine if the mean length of jail time has increased. The distribution of the population is normal.

Hypothesis testing: single mean

Is this a test of means or proportions?


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State the null and alternative hypotheses.

  • H o size 12{H rSub { size 8{O} } } {} :
  • H a size 12{H rSub { size 8{a} } } {} :

  • H o : μ = 2 . 5 size 12{H rSub { size 8{O} } :μ=2 "." 5} {} (or, H o : μ 2 . 5 size 12{H rSub { size 8{O} } :μ<= 2 "." 5} {} )
  • H a : μ > 2 . 5 size 12{H rSub { size 8{a} } :μ>2 "." 5} {}
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Is this a right-tailed, left-tailed, or two-tailed test? How do you know?


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What symbol represents the Random Variable for this test?

X ¯ size 12{ {overline {X}} } {}

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In words, define the Random Variable for this test.

The mean time spent in jail for 26 first time convicted burglars

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Is the population standard deviation known and, if so, what is it?

Yes, 1.5

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Calculate the following:

  • x ¯ = size 12{ {overline {x}} ={}} {}
  • σ = size 12{σ={}} {}
  • s x = size 12{s rSub { size 8{x} } ={}} {}
  • n = size 12{n={}} {}

  • 3
  • 1.5
  • 1.8
  • 26
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Since both σ size 12{σ} {} and s x size 12{s rSub { size 8{x} } } {} are given, which should be used? In 1 -2 complete sentences, explain why.

σ size 12{σ} {}

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State the distribution to use for the hypothesis test.

X ¯ ~ N ( 2 . 5 , 1.5 26 ) size 12{ {overline {X}} "~" N \( 2 "." 5,1 "." 5 sqrt {"26"} \) } {}

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Sketch a graph of the situation. Label the horizontal axis. Mark the hypothesized mean and the sample mean x ¯ size 12{ {overline {x}} } {} . Shade the area corresponding to the p-value.

Blank horizontal axis of the sample mean.

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Find the p-value.


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At a pre-conceived α = 0 . 05 size 12{α=0 "." "05"} {} , what is your:

  • Decision:
  • Reason for the decision:
  • Conclusion (write out in a complete sentence):

  • Reject the null hypothesis
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Discussion questions

Does it appear that the mean jail time spent for first time convicted burglars has increased? Why or why not?

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Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
1 It is estimated that 30% of all drivers have some kind of medical aid in South Africa. What is the probability that in a sample of 10 drivers: 3.1.1 Exactly 4 will have a medical aid. (8) 3.1.2 At least 2 will have a medical aid. (8) 3.1.3 More than 9 will have a medical aid.
Nerisha Reply

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Source:  OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40
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