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This is a two column model for conducting a hypotheses test for a mean with sigma known.

Step-By-Step Example of a Hypothesis Test for a Single Mean, Sigma Known (used Ex XX)

Suppose a baker claims that his bread height is more than 15 cm, on the average. Several of his customers do not believe him. To persuade his customers that he is right, the baker decides to do a hypothesis test. He randomly bakes 10 loaves of bread. The mean height of the sample loaves is 15.7 cm. The baker knows from baking hundreds of loaves of bread that the standard deviation for the height is 1 cm. and the distribution of heights is normal. Test at the 5% significance level.

State the question: State what we want to determine and what level of significance is important in your decision. We are asked to test the hypothesis that the mean bread height, μ, is more than 15 cm. We have a sample of 10 loaves. We know the population standard deviation is 1. Significance level is 5%.
Plan: Based on the above question(s) and the answer to the following questions, decide which test you will be performing.
  • Is the problem about numerical or categorical data?
  • If the data is numerical is the population standard deviation known?
  • Do you have one group or two groups?
  • What type of model do we have?
We have univariate, quantitative data. We have a sample of 10 loaves. We know the population standard deviation is 1. Therefore, we can perform a z-test (known population standard deviation). Our model will be:
X G ~ N ( μ , σ n ) = N ( 15 , 1 10 )
Hypotheses: State the null and alternative hypotheses in words then in symbolic form
  1. Express the hypothesis to be tested in symbolic form,
  2. Write a symbolic expression that must be true when the original claim is false.
  3. The null hypothesis is the statement which includes the equality.
  4. The alternative hypothesis is the statement without the equality.

Null hypothesis in words: The null hypothesis is that the true mean height of the loaves is equal to 15 cm.

Null Hypothesis symbolically: H o (Mean height) : μ = 15

Alternative Hypothesis in words: The alternative is that the true mean height on average is greater than 15 cm.

Alternative Hypothesis symbolically: H a (Mean height) : μ>15

The criteria for the inferential test stated above: Think about the assumptions and check the conditions. Summary statistics to support assumptions: If your assumptions include the need for particular types of data distribution, please indicate here and insert the appropriate graphs or charts.

Randomization Condition: The sample is a random sample.
Independence Assumption: It is reasonable to think that the loaves of bread have heights that are independent.
10% Condition: I assume the number of loaves of bread baked is more than 100, so 10 loaves is less than 10% of the population.
Sample Size Condition: Since the distribution of the bread heights is normal, my sample of 10 loaves is large enough.

Compute the test statistics: State the parameters and the sampling model The conditions are satisfied and σ is known, so we will use a hypothesis test for a mean with known standard deviation. For this calculation,we need the sample mean and standard error (SE).
x = 15.7 ; σ = 1 ; n = 10
SE = ( σ n ) = 1 10 = 0.3162
z = x - µ σ n = 15.7 - 15 0.3162 = 2.214
Determine the Critical Region(s): Based on your hypotheses are you performing a left-tailed, right tailed or two-tailed test? I will perform a right tailed test. I am only concerned about the bread being higher than 15 cm.
Sketch the test statistic and critical region: . Look up the probability on the table.

Determine the P-value

P(z<2.2134) = 0.9838; P(z>2.214) = 1-0.9838 = 0.0162

State whether you reject or fail to reject the Null hypothesis.

Since the probability is less than 2%, this is considered a rare event and the small probability leads us to reject the null hypothesis.
Conclusion: Interpret your result in the proper context, and relate it to the original question. Since the probability is less than 2%, this is considered a rare event and the small probability leads us to reject the null hypothesis. It is unlikely that a loaf of bread rises no more than 15 cm, on the average. That is, less than 2% of all loaves of bread would be at least as high as the outcome of 15.7 cm. purely by chance had the population mean height really been 15 cm. We conclude that the evidence is against the null hypothesis (the mean height is 15 cm.). There is sufficient evidence that the true mean height for the population of the baker’s loaves is greater than 15 cm.

If you reject the null hypothesis, continue to complete the following

Calculate and display your confidence interval for the Alternative hypothesis.

The mathematics for the confidence interval uses 15.7 as the mean bread height and 0.3162 as the SE. We graph a two tailed confidence interval.

x = 15.7 ; σ = 1 ; n = 10 ; SE = ( σ n ) = 1 10 = 0.3162

z-score for a two tailed test with 95% confidence is plus or minus 1.96 (read from the z-table 0.025 probability in the left tail and 0.025 probability in the right tail)

  • x - z* (SE) <µ< x + z* (SE)
  • x - 1.96 (SE) <µ< x + 1.96 (SE) ;

15.7 - 1.96 (0.3162) <µ< 15.7 + 1.96 (0.3162) ;

15.08 <µ< 16.32

Conclusion State your conclusion based on your confidence interval.

We are 95% confident that the true mean of the bakers bread height is greater than 15 cm. We are 95% confident that the population mean height is between 15.08 cm. and 16.32 cm.

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
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Source:  OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
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