Section 8 4 Choosing the sample size In estimating a population proportion how large a sample size n is necessary to achieve a particular margin of error ME Example Let s suppose we want a 4 04 ME for our 95 confidence interval We know that ME z SE so here 04 1 96 We could get our sample size by solving this equation for n but before we sample we don t know the value of So what do we do If we have a good idea from past experience what a likely value of might be we can use that If we are not sure or just want to be safe we will choose n so that we get our ME even for the largest possible value of This largest occurs when 0 5 Thus we solve 04 1 96 n for n Square both sides 04 2 1 962 Now solve for n 600 25 601 always round up to be sure we achieve our ME Thus we choose a sample size of 601 Here is our formula To insure a particular ME for a proportion choose n and round up Means In estimating a population mean how large a sample size n is necessary to achieve a particular margin of error ME Our 95 confidence interval has the form t 025 SE where SE know n and hence we don t know the df for t 025 and we don t know s Before we sample we don t So what do we do Since the t distribution is close to normal for df 30 we simply use z 1 96 as an estimate for t 025 and make a best guess for s based on experience If we call this guess then we have ME 1 96 or after squaring ME2 so that n Suppose that we want the ME for our amusement park example to be 8 What size sample should we use We will estimate from experience the standard deviation to be 25 Then 38 Thus we choose a sample size of n 38 families of 4 To insure a particular ME for a mean choose n and always round up 37 5 or
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