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Sample Size

Module by: Mary Teegarden

Summary: Calculations for determining the required sample sized when calculation a confidence interval for the population mean or population proportion.

Determining Sample Size Required to Estimate μ.

Prior to creating a confidence interval a sample must be taken. Often the number of data values needed in a sample to obtain a particular level of confidence within a given error needs to be determined prior to taking the sample. If the sample is too small the result may not be useful and if the sample is too big both time and money are wasted in the sampling.

From the formula for the error bound, the following formula can be derived:

Sample Size for Estimating Mean μ

n = z α / 2 σ E 2 n = z α / 2 σ E 2 size 12{n= left [ { {z rSub { size 8{ {α} slash {2} } } σ} over {E} } right ] rSup { size 8{2} } } {} (1)
  • Where zα/2zα/2 size 12{z rSub { size 8{ {α} slash {2} } } } {} = the critical z score based on the desired confidence level
  • E = desired margin of error
  • σ = population standard deviation

Often the population standard deviation is unknown. Often the sample standard deviation from a previous sample of size greater than 30 may be used as an approximation to σ.

Round Off Rule for Sample Size n

Often times the value found by using the formula for sample size is not a whole number. However the sample size must be a whole number, so always round up to the next larger whole number.

Example

Suppose the scores on a statistics final are normally distributed with a standard deviation of 10 points. You have been asked to construct a 95% confidence interval with an error of no more than 2 points.

z0.25z0.25 size 12{z rSub { size 8{0 "." "25"} } } {} = 1.645

E = 2

σ = 10

n=zα/2σE2=(1.645)(10)22n=zα/2σE2=(1.645)(10)22 size 12{n= left [ { {z rSub { size 8{ {α} slash {2} } } σ} over {E} } right ] rSup { size 8{2} } = left [ { { \( 1 "." "645" \) \( "10" \) } over {2} } right ] rSup { size 8{2} } } {}= 67.6506

Hence, a sample of size 68, must be taken to create a 95% confidence interval with an error of no more than two points.

Determining Sample Size Required to Estimate p.

To determine the sample size necessary to ensure a given error for a particular confidence level, the formula for the error bound can be rewritten as follows:

n = z α / 2 E 2 p ( 1 − p ) n = z α / 2 E 2 p ( 1 − p ) size 12{n= left [ { {z rSub { size 8{ {α} slash {2} } } } over {E} } right ] rSup { size 8{2} } p \( 1 - p \) } {} (2)
  • Where zα/2zα/2 size 12{z rSub { size 8{ {α} slash {2} } } } {} = the critical z score based on the desired confidence level
  • E = desired margin of error
  • p = population proportion

Generally the population proportion is unknown and p’ is determined using a previous sample. Hence

n = z α / 2 E 2 p ' ( 1 − p ' ) n = z α / 2 E 2 p ' ( 1 − p ' ) size 12{n= left [ { {z rSub { size 8{ {α} slash {2} } } } over {E} } right ] rSup { size 8{2} } p' \( 1 - p' \) } {} (3)

If there is no previous sample then p = 0.5 is used since it maximized the value of p(1 - p). Hence

n = z α / 2 E 2 0 . 5 ( 1 − 0 . 5 ) = z α / 2 E 2 0 . 25 n = z α / 2 E 2 0 . 5 ( 1 − 0 . 5 ) = z α / 2 E 2 0 . 25 size 12{n= left [ { {z rSub { size 8{ {α} slash {2} } } } over {E} } right ] rSup { size 8{2} } 0 "." 5 \( 1 - 0 "." 5 \) = left [ { {z rSub { size 8{ {α} slash {2} } } } over {E} } right ] rSup { size 8{2} } 0 "." "25"} {} (4)

Suppose Halmark wish to know what proportion of oldest children buy their mothers a Mother’s Day Card. (See example 8 -5) How many people must be sampled is they wish to be 95% certain that the proportion is within 2%?

a) Use the following sample data as an estimate for the population proportion.

Given that 421 of 500 responded in the affirmative, p’ = 421500421500 size 12{ { {"421"} over {"500"} } } {}= 0.842

z0.25z0.25 size 12{z rSub { size 8{0 "." "25"} } } {} = 1.645

E = 0.02

p’ = 0.842

n=1.6450.0220.842(0.158)n=1.6450.0220.842(0.158) size 12{n= left [ { {1 "." "645"} over {0 "." "02"} } right ] rSup { size 8{2} } 0 "." "842" \( 0 "." "158" \) } {}= 899.997

Hence 900 people need to be surveyed to ensure a 95% confidence interval with an error of at most 2%.

b) Suppose there is no previous sample. How many people need to be surveyed?

z0.25z0.25 size 12{z rSub { size 8{0 "." "25"} } } {} = 1.645

E = 0.02

assume p = 0.5

n=1.6450.0220.25n=1.6450.0220.25 size 12{n= left [ { {1 "." "645"} over {0 "." "02"} } right ] rSup { size 8{2} } 0 "." "25"} {}= 1691.27

Hence 1692 people need to be surveyed to ensure a 95% confidence interval with an error of at most 2%.

Note that not having a previous sample greatly increases the number of data values needed in a sample. Often a pilot study is done to generate an approximation for p.

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