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However, in the error bound formula, we use p ' q ' n as the standard deviation, instead of p q n

In the error bound formula, the sample proportions p ' and q ' are estimates of the unknown population proportions p and q . The estimated proportions p ' and q ' are used because p and q are not known. p ' and q ' are calculated from the data. p ' is the estimated proportion of successes. q ' is the estimated proportion of failures.

For the normal distribution of proportions, the z-score formula is as follows.

If P ' ~ N ( p , p q n ) then the z-score formula is z = p ' - p p q n

Suppose that a market research firm is hired to estimate the percent of adults living in a large city who have cell phones. 500 randomly selected adult residents this city are surveyed to determine whether they have cell phones. Of the 500 people surveyed, 421 responded yes - they own cell phones. Using a 95% confidence level, compute a confidence interval estimate for the trueproportion of adults residents of this city who have cell phones.

Let X = the number of people in the sample who have cell phones. X is binomial. X ~ B ( 500 , 421 500 ) .

To calculate the confidence interval, you must find p ' , q ' , and EBP .

n = 500 x = the number of successes = 421

p ' = x n = 421 500 = 0.842

p ' = 0.842 is the sample proportion; this is the point estimate of the population proportion.

q ' = 1 - p ' = 1 - 0.842 = 0.158

Since CL = 0.95 , then α = 1 - CL = 1 - 0.95 = 0.05 α 2 = 0.025 .

z α 2 = z .025 = 1.96

Use the TI-83, 83+ or 84+ calculator command invnorm(.975,0,1) to find z .025 . Remember that the area to the right of z .025 is 0.025 and the area to the left of z .025 is 0.975. This can also be found using appropriate commands on other calculators, using a computer, or using a Standard Normal probability table.

EBP = z α 2 p ' q ' n = 1.96 [ ( .842 ) ( .158 ) 500 ] = 0.032

p ' - EBP = 0.842 - 0.032 = 0.81

p ' + EBP = 0.842 + 0.032 = 0.874

The confidence interval for the true binomial population proportion is ( p ' - EBP , p ' + EBP ) = ( 0.810 , 0.874 ) .


We estimate with 95% confidence that between 81% and 87.4% of all adult residents of this city have cell phones.

Explanation of 95% confidence level

95% of the confidence intervals constructed in this way would contain the true value for the population proportion of all adult residents of this city who have cell phones.

For a class project, a political science student at a large university wants to determine the percent of students that are registered voters. He surveys 500students and finds that 300 are registered voters. Compute a 90% confidence interval for the true percent of students that are registered voters and interpret the confidenceinterval.

x = 300 and n = 500 . Using a TI-83+ or 84 calculator, the 90% confidence interval for the true percent of students that are registered voters is (0.564, 0.636).

p ' = x n = 300 500 = 0.600

q ' = 1 - p ' = 1 - 0.600 = 0.400

Since CL = 0.90 , then α = 1 - CL = 1 - 0.90 = 0.10 α 2 = 0.05 .

z α 2 = z .05 = 1.645

Use the TI-83, 83+ or 84+ calculator command invnorm(.95,0,1) to find z .05 . Remember that the area to the right of z .05 is 0.05 and the area to the left of z .05 is 0.95. This can also be found using appropriate commands on other calculators, using a computer, or using a Standard Normal probability table.

EBP = z α 2 p ' q ' n = 1.645 [ ( .60 ) ( .40 ) 500 ] = 0.036

p ' - EBP = 0.60 - 0.036 = 0.564

p ' + EBP = 0.60 + 0.036 = 0.636


  • We estimate with 90% confidence that the true percent of all students that are registered voters is between 56.4% and 63.6%.
  • Alternate Wording: We estimate with 90% confidence that between 56.4% and 63.6% of ALL students are registered voters.

Explanation of 90% confidence level

90% of all confidence intervals constructed in this way contain the true value for the population percent of students that are registered voters.

Calculating the sample size

If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.

The error bound formula for a proportion is EBP = z α 2 p ' q ' n . Solving for n gives you an equation for the sample size:

n z 2 p' q' EBP 2 , where z = z α 2

Suppose a mobile phone company wants to determine the current percentage of customers aged 50+ that use text messaging on their cell phone. How many customers aged 50+ should the company survey in order to be 90% confident that the estimated (sample) proportion is within 3 percentage points of the true population proportion of customers aged 50+ that use text messaging on their cell phone.

From the problem, we know that EBP=0.03 (3%=0.03) and z α 2 = z .05 = 1.645 because the confidence level is 90%

However, in order to find n , we need to know the estimated (sample) proportion p'. Remember that q'=1-p'. But, we do not know p' yet. Since we multiply p' and q' together, we make them both equal to 0.5 because p'q'= (.5)(.5)=.25 results in the largest possible product. (Try other products: (.6)(.4)=.24; (.3)(.7)=.21; (.2)(.8)=.16 and so on). The largest possible product gives us the largest n. This gives us a large enough sample so that we can be 90% confident that we are within 3 percentage points of the true population proportion. To calculate the sample size n, use the formula and make the substitutions.

n z 2 p' q' EBP 2 gives n 1.645 2 (.5) (.5) .03 2 =751.7

Round the answer to the next higher value. The sample size should be 758 cell phone customers aged 50+ in order to be 90% confident that the estimated (sample) proportion is within 3 percentage points of the true population proportion of all customers aged 50+ that use text messaging on their cell phone.

Questions & Answers

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Sanjika Reply
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mCQ creativity involves
ology means a study of
SharaAmor Reply
is a greek word means soul
a branch of knowledge
psych means soul and ology means to study.
Can someone help me with tha latent functions of different social institutions
***interestinglydifferenttopics.blogspot.com/2020/10/reflexes-psychological-debate.html Do check this out and please correct me if I'm wrong anywhere in this
Interestingly Reply
Factors that affect learning?
Ofentse Reply
I am struggling with survivors guilt and complicated grief after traumatic death of spouse. how to cope.
Rowdy Reply
Try to talk with someone you trust and join people who makes you feel worthy. try to help someone. Helping someone makes you feel better
I think you must seek professional help.
I am extremely sorry for your loss...I know that the pain of your loss is overwhelming and that you are experiencing all kinds of difficult and unexpected emotions. please don't get me wrong but the best way to cope is to move on..i know its hard..but that's the only way..(continued.)
stop burdening yourself with thoughts that say you could have done something..you couldn't..your just a human being and you cant always be a superman who saves the day..please dont feel guilty when you have no reason to..(cont.)
learn to love yourself again..learn to live ..for yourself,for the people who love you and need you in their lives.live for people who you once loved...Think about how you can make your life meaningful once again.It might seem impossible but trust me its necessary ..
Let your feelings of pain out ..you dont have to suffer ..talk to people whom you trust and love. take care and have a beautiful life ahead.
First and foremost -- Wow Rowdy! You have been through a highly difficult situation in your life, hands down. As a wife, I simply cannot phathom nor imagine slightly the feelings and thoughts you bare and I'm sure as these words do not help and hit rewind; my heart and prayers are going out to you-
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yourself, which in turn could make you question who you are right now and what this means for you as a person now and in the future. This is all okay, but there will have to be an understanding as to how to grow from these feelings and thoughts in a more humane manner l as these feelings and ---
thoughts could take a turn for a much more negative outcome as you find yourself trapped in a darkness that, if strong enough, trials and tribulations take years of complex development and experience, less strong minded individuals sometimes either never find the light, or end up being stuck in ----
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to guide you through; keep an your senses wide open so that you can learn when she is trying to communicate.
What is meant by politics of intelligence? Please guide about this article.
Ali Reply
Thanks angela
Can u please explain it to me in your words?
Information about child psychology
Mohammed Reply
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help me please:<
actually I don't know
I badly need some for my research :<
But I need that😥
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branches of psychology
QSN Reply
Legal psychology
Developmental psychology, Sports psychology , Clinical psychology , Marketing psychology , counseling psychology , Biological psychology, Educational psychology , Positive psychology etc.
social, occupational psychology
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John Reply
the cortex refers to what part of the brain
the cortex refers to what part of the brain
cerebral cortex
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marjelee Reply
during bad experience
i think its not about suitation, its about the thing tatz need training.
training is possibly warranted for all situations.
I would think that training should be an ungoing thing. That's before, during and after. just in case there's a reoccurrence
any time
what is Karma
ana Reply
I think the summation of good and bad deeds you do in your life. Accidentally or otherwise.
Yes. I totally agree.
And if it is bad then the root must have been too.
Yes , In Hinduism we also believe that our parent's Karma also applies to us ! Don't know about that , but it kept people in check .
Adding to Swarada point. karma can be good or bad. both will have consequences. karma is biasedness you have shown in your life. like a king could not have two rules for normal man and his son, that comes under karma.
Oh perfectly said!
Ok. It is also said that when one does something... bad or good..it will come back to you 10 folds of the same.
yea that too ! I remember my mother always said if u stole someone's 10 rupees ur 100 will be stolen ( excuse my English ! I'm a non - English speaking person)
karma is nothing but what one usually mean when they say- what you sow, so shall you reap.
more important.. this concept was told by lord Krishna as part of spirituality, to focus on your good deed ,the purpose you are born for and your responsibilities. Not for hoping that bad will happen with people who made you feel bad. just giving example ;)
karma is the boomerang of your actions. what u put out in the universe u get back
Who is the founder of psychology
Daryl Reply
Wilhelm wundt
Wilhelm Wundt
William wundth
Wilhelm Wundt
and William James
Wilhelm Wundt
Wilhelm Wundt is the founder of structuralism
Yes and also the "Father of Psychology" I think the correct word is Father not Founder.
oh ok. noted
i believe it goes beyond him I believe it is Ancient but un formalized
I like your answer Swarada
Oh thanks! But u summarised it well!
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what are some academic research on developmental psychology
Sawrada Founder of Psychology is W.Wundt but Father of Psychology is Sigmund Freud...
what are some good schools to study psychology?
Enjoy Reply
George Mason, Penn State, UVA, university of Michigan
Hi I'm currently doing my experimental psychology research can you suggest a topic or research topic for my research thank you and please see 😊
Jerico Reply
Hello Jerico. I could help you out with some ideas if you're interested!
can u give me some ideas too?
For sure. It depends if you're trying to do an experiment (using statistics probably) or a case study what are you aiming for?
Hello Jerico. Try to opt for something simple and easy to do. I wouls say perhaps study the effects of color on concentration? You could get a group of people and have them take a memory test (you can find some online) in a white room for example. Then have them take another memory test
similar to the first one but in a different room and then you check the responses if the performance is better
but you have to know that a lot of factors will interfere in the performance such as mental state etc so make sure to choose a certain specific population. that's very important
you could be a bit more specific about it , to isolate the variables that can mess it up like age, sex, etc... you gotta take that into account
so that the results get somewhat accurate
that is very detailed information I really like it thank you very much 😊 your so very kind
You're very welcome
what are your interests in general?
Althogh I'm persuing botany and engineering.... but I also want to study human behaviour and psychology.... I would like if anyone can share their knowledge with me !!!!
I'm interested on effect of electroconvulsive therapy on people having a depression
What is Electroconvulsive therapy?
Astral Projection
Electroconvulsive therapy (ECT) is a medical treatment that is most commonly used in patients with severe major depression.
What are the expected dutties functions and responsibilities of a head of an agency based on the principles, Practice of industrial psychology?
Paul Reply
what are the psychological characteristics expected from an employee wether regular or occupying a position?
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