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,
[tex]\sigma = 3.10\text{ milligrams}[/tex]
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:
[tex]z_{critical}\times \dfrac{\sigma}{\sqrt{n}}[/tex]
[tex]z_{critical}\text{ at}~\alpha_{0.05} = 1.96[/tex]
Putting values, we get,
[tex]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[/tex]
Thus, the minimum sample size required to construct a 95% confidence interval for the population mean is 65.
