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

The formula for calculating margin of error is given to be

`\begin{gathered} E=0.4 \\ n=? \\ \sigma=4.8 \\ p(x<(Z_(\alpha))/(2))=(1-0.9)/(2)=(0.1)/(2)=0.05 \\ From\text{ the z-score and probability converter table, } \\ (Z_(\alpha))/(2)=1.645 \end{gathered}`
`\begin{gathered} Thus, \\ 0.4=1.645((4.8)/(√(n))) \\ Divide\text{ both sides by 1.645} \\ (0.4)/(1.645)=(4.8)/(√(n)) \\ 0.24316=(4.8)/(√(n)) \\ 0.24316√(n)=4.8 \\ √(n)=(4.8)/(0.24316) \\ √(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)