The practice questions here help you in determining appropriate sample sizes required to achieve a certain margin of error. Figure out the sample size needed in the following problems.

## Sample questions

A physician wants to estimate the average BMI (

*body mass index,*a measure that combines information about height and weight) for her adult patients.She decides to draw a sample of clinical records and retrieve this information from them. She wants an estimate with a margin of error of 1.5 units of BMI, with 95% confidence, and believes that the national population standard deviation of adult BMI of 4.5 also applies to her patients. She knows that BMI is approximately normally distributed for adults. How large a sample does she need to draw?

**Answer:**The sample size must be at least 35 to produce a margin of error of plus/minus 1.5 units of BMI.The formula to calculate the sample size needed for a confidence interval for a sample mean is

where

*n*is the sample size required,*z**is the value from the table for the chosen confidence level,is the population standard deviation, and MOE is the margin of error. For this example,

*z**is 1.96,is 1.6, and the MOE is 1.5.

Samples sizes are always rounded up to the nearest integer, so the sample size must be at least 35.

A physician wants to estimate the height of 6-year-old boys in her community, using a random sample drawn from administrative records.

She wants an estimate with a margin of error of 0.5 inches, with 95% confidence, and believes that the population standard deviation of 1.8 inches applies to her population. She also knows that height is approximately normally distributed for this population. How large a sample does she need to draw?

**Answer:**The sample size must be at least 50 to produce a margin of error of plus/minus 0.5 inches.The formula to calculate sample size to estimate a mean, when the standard deviation is known, is

where

*n*is the sample size required,*z**is the value from the table for the chosen confidence level,is the population standard deviation, and MOE is the margin of error. For this example,

*z**is 1.96,is 1.6, and the MOE is 0.5.

She needs a sample of at least 50 to achieve the desired precision.

You want to estimate the average height of 10-year-old boys in your community.

The population standard deviation is 3 inches. What size sample do you need for a margin of error of no more than plus/minus 1 inch and a confidence level of 95% when constructing a confidence interval for the mean height of all 10-year-old boys?

**Answer:**The sample size must be at least 35 to produce a margin of error of plus/minus 1 inch.The formula to find the required sample size based on a desired margin of error is

Here, MOE is the margin of error,

*z** is the*z**-value corresponding to your desired confidence level, andis the population standard deviation. If

is unknown, you can do a small pilot study to find the standard deviation of the sample (including making a conservative adjustment to the sample standard deviation to be safe).

Substitute the known values into the formula:

Always round up the answer to the nearest whole number to be sure the sample size is large enough to give the margin of error needed. So

*n*greater than or equal to 35. That means that you need at least 35 boys in your sample to get a margin of error of no more than 1 inch for average height.Note that a sample size of 35 will give you the margin of error you want; a higher sample size will give you an even lower margin of error.

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