Question

Suppose we wish to design a study to investigate the effects of loud music on teenagers’ ability
to concentrate. We know from previous studies that the standard deviation of time to complete
this task is 3.4 minutes. How many subjects would be required to ensure with 90% confidence
that the generated estimate is within 1 minute of the true mean time required?

Answer 1

Let $m$ denote the margin of error, then $m$ and it is given by,

$m=Z_{\alpha/2}*(\sigma/\sqrt{n})$

In order to determine the number of subjects required, make n subject of the formula,

Therefore,

$m*\sqrt{n}=Z_{\alpha/2}*\sigma$

$\sqrt{n}=(Z_{\alpha/2}*\sigma)/m$

Hence

$n=((Z_{\alpha/2}*\sigma)/m)^2$

Values given are,

$m=1$

$\sigma=3.4$

$Z_{\alpha/2}=Z_{0.1/2}=Z_{0.05}=1.645$

$n$ therefore is,

$n=((1.645*3.4)/1)^2=(5.593)^2=31.281649\approxeq32subjects$

Therefore $n=32$ subjects would be required to ensure with 90% confidence

that the generated estimate is within 1 minute of the true mean time.