<< Chapter < Page Chapter >> Page >
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

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
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 ?
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
can nanotechnology change the direction of the face of the world
Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Collaborative statistics using spreadsheets' conversation and receive update notifications?