10 - Determining the Sample Size Necessary for a Desired ...



11 - Determining the Sample Size Necessary to Obtain a Desired Margin of Error

Population Mean (μ)

In the handout 9 – Sampling Distributions we found that the interval

[pic] up to [pic]

had a 95% chance of covering the population mean. The margin of error for this interval is

[pic]

If we wanted our margin of error to be at most E units what sample size should we use?

This says that to obtain a 95% CI for μ with a margin of error no larger than E we should use a sample size of

[pic]

However we cannot calculate this in practice unless we know [pic]. Which of course we don’t and furthermore we don’t even know s, the sample standard deviation, until we have our data in hand. Thus in order to use this result we need to plug in a “best guess” for [pic]. This guess might come from:

• Pilot study where s = sample standard deviation is calculated

• Prior studies (literature reviews)

• Use approximation based on the Range, [pic]. Granted we don’t the range until the data is collected, but we might be able to guess the largest and smallest values we might expect to see when collect our data.

• In general, using a[pic]which is too large is better than using one that is too small.

Example: What sample size would be necessary to estimate the mean protein content found in zebra mussels with a 95% confidence interval that has a margin of error no larger than 1 mg/g?

Population Proportion (p)

In the handout 9 – Sampling Distributions we found that the interval

[pic] up to [pic]

had a 95% chance of covering the population proportion. The margin of error for this interval is

[pic]

If we wanted this to be at most E units what sample size should we use?

This says that to obtain a 95% CI for p with a margin of error no larger than E we should use a sample size of

[pic]

However we cannot calculate this in practice unless we know p. Which of course we don’t and furthermore we don’t even know[pic], the sample proportion, until we have our data in hand. In order to use this result we need to plug in a “best guess” for p. This guess might come from:

• Pilot study where [pic]= sample proportion is calculated

• Prior studies (literature reviews)

• Use the worst case scenario by noting that [pic]and is equal to .25 when p=.50. Using p = .50 simplifies the formula to

[pic]

If you have no “best guess” for p this conservative approach is the one

you should take.

Example: How many trout would need to be caught and released to estimate the mortality rate when barbless hooks are used, if researchers initially believe the mortality rate is no larger than 15% and wish to estimate the true mortality rate using a 95% confidence interval with a margin of error no larger than E = .03?

What if they wish to assume nothing about the mortality rate initially?

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