Question

bacteria reveals a sample mean of ¯x = 70 hours with a standard deviation of s = 4.8 hours.What sample size should you gather to achieve a 0.4 hour margin of error? Round your answer up to the nearest whole number.

Answer 1

Solution

The formula for calculating margin of error is given to be

$\begin{gathered} E=0.4 \\ n=? \\ \sigma=4.8 \\ p(x<\frac{Z_{\alpha}}{2})=\frac{1-0.9}{2}=\frac{0.1}{2}=0.05 \\ From\text{ the z-score and probability converter table, } \\ \frac{Z_{\alpha}}{2}=1.645 \end{gathered}$$\begin{gathered} Thus, \\ 0.4=1.645(\frac{4.8}{\sqrt{n}}) \\ Divide\text{ both sides by 1.645} \\ \frac{0.4}{1.645}=\frac{4.8}{\sqrt{n}} \\ 0.24316=\frac{4.8}{\sqrt{n}} \\ 0.24316\sqrt{n}=4.8 \\ \sqrt{n}=\frac{4.8}{0.24316} \\ \sqrt{n}=19.74 \\ n=19.74^2 \\ n=389.67 \\ n=390(nearest\text{ whole number\rparen} \end{gathered}$

n = 390 bacteria (nearest whole number)