Answer:
The minimum sample size required to construct a 95% confidence interval for the population mean is 65.
Step-by-step explanation:
We are given the following in the question:
Population standard deviation,
![\sigma = 3.10\text{ milligrams}](https://tex.z-dn.net/?f=%5Csigma%20%3D%203.10%5Ctext%7B%20milligrams%7D)
We need to construct a 95% confidence interval such that the estimate is within 0.75 milligrams of the population mean.
Thus, the margin of error must me 0.75
Formula for margin of error:
![z_{critical}\times \dfrac{\sigma}{\sqrt{n}}](https://tex.z-dn.net/?f=z_%7Bcritical%7D%5Ctimes%20%5Cdfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D)
![z_{critical}\text{ at}~\alpha_{0.05} = 1.96](https://tex.z-dn.net/?f=z_%7Bcritical%7D%5Ctext%7B%20at%7D~%5Calpha_%7B0.05%7D%20%3D%201.96)
Putting values, we get,
![0.75 = 1.86\times \dfrac{3.10}{\sqrt{n}}\\\\\sqrt{n} = \dfrac{1.96\times 3.10}{0.75}\\\\\sqrt{n} = 8.101\\\Rightarrow n = 65.63\approx 65](https://tex.z-dn.net/?f=0.75%20%3D%201.86%5Ctimes%20%5Cdfrac%7B3.10%7D%7B%5Csqrt%7Bn%7D%7D%5C%5C%5C%5C%5Csqrt%7Bn%7D%20%3D%20%5Cdfrac%7B1.96%5Ctimes%203.10%7D%7B0.75%7D%5C%5C%5C%5C%5Csqrt%7Bn%7D%20%3D%208.101%5C%5C%5CRightarrow%20n%20%3D%2065.63%5Capprox%2065)
Thus, the minimum sample size required to construct a 95% confidence interval for the population mean is 65.