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Calculating the sample size n

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 population proportion is

  • E B P = ( z α 2 ) ( p q n )
  • Solving for n gives you an equation for the sample size.
  • n = ( z α 2 ) 2 ( p q ) E B P 2

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

From the problem, we know that EBP = 0.03 (3%=0.03) and z α 2 z 0.05 = 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 ′ = (0.5)(0.5) = 0.25 results in the largest possible product. (Try other products: (0.6)(0.4) = 0.24; (0.3)(0.7) = 0.21; (0.2)(0.8) = 0.16 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 three 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 E B P 2 gives n = 1.645 2 ( 0.5 ) ( 0.5 ) 0.03 2 = 751.7

Round the answer to the next higher value. The sample size should be 752 cell phone customers aged 50+ in order to be 90% confident that the estimated (sample) proportion is within three percentage points of the true population proportion of all customers aged 50+ who use text messaging on their cell phones.

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Try it

Suppose an internet marketing company wants to determine the current percentage of customers who click on ads on their smartphones. How many customers should the company survey in order to be 90% confident that the estimated proportion is within five percentage points of the true population proportion of customers who click on ads on their smartphones?

271 customers should be surveyed.Check the Real Estate section in your local

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References

Jensen, Tom. “Democrats, Republicans Divided on Opinion of Music Icons.” Public Policy Polling. Available online at http://www.publicpolicypolling.com/Day2MusicPoll.pdf (accessed July 2, 2013).

Madden, Mary, Amanda Lenhart, Sandra Coresi, Urs Gasser, Maeve Duggan, Aaron Smith, and Meredith Beaton. “Teens, Social Media, and Privacy.” PewInternet, 2013. Available online at http://www.pewinternet.org/Reports/2013/Teens-Social-Media-And-Privacy.aspx (accessed July 2, 2013).

Prince Survey Research Associates International. “2013 Teen and Privacy Management Survey.” Pew Research Center: Internet and American Life Project. Available online at http://www.pewinternet.org/~/media//Files/Questionnaire/2013/Methods%20and%20Questions_Teens%20and%20Social%20Media.pdf (accessed July 2, 2013).

Questions & Answers

How do you get log of normal population
Shan Reply
plz give me answer of this question show that mean of population or sampling distribution are equally
Mari
what is the probability of getting no head face up in three tosses of a fair coin?
Epara Reply
In how many ways can probability be assigned to an event of interest?
Epara
Hey guys can someone help me with combinations and permutations
Lion
Phone lines on an airline system are occupied 50% of time assume that 10 calls are placed to the airline. What is the probability at least 1 call the lines are occupied?
Highness Reply
Phone lines on an airline system are occupied 50% of time assume that 10 calls are placed to the airline. What is the probability at least 1 call the lines are occupied?
Highness Reply
Why is the method of selecting the sample even more important than the size of the sample?
Nana Reply
formulaas for gruoped and ungrouped of quartiles
chatered Reply
Why is the method of selecting the sample even more important than the size of the sample?
Nana
una empresa productora que participa en el mercado de lapices tiene la siguente funcion de demanda Qd=a la raiz de 9 -9p si la elasticidad precio de demanda de la empresa es epd=0,5 determinar a) hallar el precio y cantidad
Jany Reply
find the mean mew and the standard devation sigma of the given population 9.8,10.2,10.4,9.8,10.0,10.2,9.6
Vaneza Reply
1. A card is drawn at random from an ordinary deck of 52 playing cards.* *Find the probability that it is* (a). an ace (b). a jack of hearts (c). a three of clubs or a six of diamonds (d). a heart (e). any suit except hearts (f). a ten or a spade (g). neither a four nor a club. *Hint:* 1st determine how many of the following is in a deck of cards. A deck of cards have 52 cards ♠️- Club ♣️ - Spade ♥️ - Heart ♦️- Diamond
Agness Reply
a measure of central tendency is a typical value around which other figures congregate
Anand Reply
hmm mean mode median
Umar
nothing
farri
Y = alpha0 + alpha1X1 + E what is this equation
Musawenkosi Reply
An estimated linear regression equation
Carlos
Thank you
Musawenkosi
simple linear regression .. where Alpho zero is reg constant ( intercept of the reg line) and Alpha1 is the regression coefficient ( slope of the regression line)
Umar
null and altarnate hypothesis are the statement about
farri Reply
about any population of interest
Umar
don't know
farri
I said we give hypothesis about any population and mean , in null hyp we say sample mean is equal to the population mean where as in alternate hyp we say sample mean is not equal to the pop mean .. to test these things we use students test statistics commonly
Umar
ok thnx
farri
welcome
Umar
probability of getting one black card
kaynat Reply
from a standard deck?
Umar
it will be 1/2
Umar
coz there are 26 blacks card out of 52 in a deck
Umar
so prob of getting a black card out of the deck = 26/52 = 1/5 =0.5
Umar
when are not two events mutually exclusive? 1) they overlap in a venn diagram 2) Probability of one affects the other
Junaid Reply
both
Umar
A stenographer claims that she can take dictation at the rate of 120 words per minute can we reject her claim on the basis of 100 trails in which she demonstrates a mean of 116 words with a variance of 225 words
Aromal Reply
the hypothesis to be tested is the claim to be tested
smita
H0= 120,Ha not equal to 120 x bar =116,s=225,n=100
smita

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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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