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A fisherman is interested in whether the distribution of fish caught in Green Valley Lake is the same as the distribution of fish caught in Echo Lake. Of the 191 randomly selected fish caught in Green Valley Lake, 105 were rainbow trout, 27 were other trout, 35 were bass, and 24 were catfish. Of the 293 randomly selected fish caught in Echo Lake, 115 were rainbow trout, 58 were other trout, 67 were bass, and 53 were catfish. Perform the hypothesis test at a 5% level of significance.

  • 3
  • Chi-Square with df = 3
  • 11.75
  • p-value = 0.0083

  • ii. Reject the null hypothesis.
    iv. There is sufficient evidence to conclude that the distribution of fish in Green Valley Lake is not the same as the distribution of fish in Echo Lake.

A plant manager is concerned her equipment may need recalibrating. It seems that the actual weight of the 15 oz. cereal boxes it fills has been fluctuating. The standard deviation should be at most 1 2 size 12{ { { size 8{1} } over { size 8{2} } } } {} oz. In order to determine if the machine needs to be recalibrated, 84 randomly selected boxes of cereal from the next day’s production were weighed. The standard deviation of the 84 boxes was 0.54. Does the machine need to be recalibrated?

  • 83
  • Chi-Square with df = 83
  • 96.81
  • p-value = 0.1426; There is a 0.1426 probability that the sample standard deviation is 0.54 or more.
  • Decision: Do not reject null; Conclusion: There is insufficient evidence to conclude that the standard deviation is more than 0.5 oz. It cannot be determined whether the equipment needs to be recalibrated or not.

Consumers may be interested in whether the cost of a particular calculator varies from store to store. Based on surveying 43 stores, which yielded a sample mean of $84 and a sample standard deviation of $12, test the claim that the standard deviation is greater than $15.

Isabella, an accomplished Bay to Breakers runner, claims that the standard deviation for her time to run the 7 ½ mile race is at most 3 minutes. To test her claim, Rupinder looks up 5 of her race times. They are 55 minutes, 61 minutes, 58 minutes, 63 minutes, and 57 minutes.

  • 4
  • Chi-Square with df = 4
  • 4.52
  • 0.3402
  • Decision: Do not reject null.

Airline companies are interested in the consistency of the number of babies on each flight, so that they have adequate safety equipment. They are also interested in the variation of the number of babies. Suppose that an airline executive believes the average number of babies on flights is 6 with a variance of 9 at most. The airline conducts a survey. The results of the 18 flights surveyed give a sample average of 6.4 with a sample standard deviation of 3.9. Conduct a hypothesis test of the airline executive’s belief.

The number of births per woman in China is 1.6 down from 5.91 in 1966 (Source World Bank, 6/5/12 ). This fertility rate has been attributed to the law passed in 1979 restricting births to one per woman. Suppose that a group of students studied whether or not the standard deviation of births per woman was greater than 0.75. They asked 50 women across China the number of births they had. Below are the results. Does the students’ survey indicate that the standard deviation is greater than 0.75?

# of births Frequency
0 5
1 30
2 10
3 5
  • 49
  • Chi-Square with df = 49
  • 54.37
  • p-value = 0.2774; If the null hypothesis is true, there is a 0.2774 probability that the sample standard deviation is 0.79 or more.
  • Decision: Do not reject null; Conclusion: There is insufficient evidence to conclude that the standard deviation is more than 0.75. It cannot be determined if the standard deviation is greater than 0.75 or not.

According to an avid aquariest, the average number of fish in a 20–gallon tank is 10, with a standard deviation of 2. His friend, also an aquariest, does not believe that the standard deviation is 2. She counts the number of fish in 15 other 20–gallon tanks. Based on the results that follow, do you think that the standard deviation is different from 2? Data: 11; 10; 9; 10; 10; 11; 11; 10; 12; 9; 7; 9; 11; 10; 11

The manager of "Frenchies" is concerned that patrons are not consistently receiving the same amount of French fries with each order. The chef claims that the standard deviation for a 10–ounce order of fries is at most 1.5 oz., but the manager thinks that it may be higher. He randomly weighs 49 orders of fries, which yields a mean of 11 oz. and a standard deviation of 2 oz.

  • σ 2 1 . 5 2 size 12{σ rSup { size 8{2} }<= left (1 "." 5 right ) rSup { size 8{2} } } {}
  • 48
  • Chi-Square with df = 48
  • 85.33
  • 0.0007
  • Decision: Reject null.

Try these true/false questions.

As the degrees of freedom increase, the graph of the chi-square distribution looks more and more symmetrical.


The standard deviation of the chi-square distribution is twice the mean.


The mean and the median of the chi-square distribution are the same if df = 24 size 12{ ital "df"="24"} {} .


In a Goodness-of-Fit test, the expected values are the values we would expect if the null hypothesis were true.


In general, if the observed values and expected values of a Goodness-of-Fit test are not close together, then the test statistic can get very large and on a graph will be way out in the right tail.


The degrees of freedom for a Test for Independence are equal to the sample size minus 1.


Use a Goodness-of-Fit test to determine if high school principals believe that students are absent equally during the week or not.


The Test for Independence uses tables of observed and expected data values.


The test to use when determining if the college or university a student chooses to attend is related to his/her socioeconomic status is a Test for Independence.


The test to use to determine if a six-sided die is fair is a Goodness-of-Fit test.


In a Test of Independence, the expected number is equal to the row total multiplied by the column total divided by the total surveyed.


In a Goodness-of Fit test, if the p-value is 0.0113, in general, do not reject the null hypothesis.


For a Chi-Square distribution with degrees of freedom of 17, the probability that a value is greater than 20 is 0.7258.


If df = 2 size 12{ ital "df"=2} {} , the chi-square distribution has a shape that reminds us of the exponential.


Questions & Answers

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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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?
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Berger describes sociologists as concerned with
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Source:  OpenStax, Collaborative statistics for mt230. OpenStax CNX. Aug 18, 2011 Download for free at http://legacy.cnx.org/content/col11345/1.2
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