You are a researcher studying the lifespan of a certain species of bacteria. (Thanks for the help!)

Question

A preliminary sample of 30 bacteria reveals a sample mean of ¯ x = 70 hours with a standard deviation of s = 4.6 hours. You would like to estimate the mean lifespan for this species of bacteria to within a margin of error of 0.7 hours at a 90% level of confidence.

What sample size should you gather to achieve a 0.7 hour margin of error? Round your answer up to the nearest whole number.

n = ___________ bacteria

Answer 1

The z score for 90% confidence level is 1.645 You can find this with a z table or a calculator

From the empirical rule you know it has to be somewhere between 1 and 2.

Take the z score & multiply it times the standard deviation, 1.645 x 4.6 = 7.567

7.567/sqrN = 0.7 solve for N, the sample size

cross multiply

0.7sqrN = 7.567

sqrN = 7.567/0.7 = 5.2969

square both sides

N = 5.2969 squared = 28.057

Round up to to 29

The confidence interval with 90% level is

mean plus or minus z score times standard deviation divided by square root of sample size

X = 70 + or - (1.645)(4.6)/N^1/2 Set the 1.645(4.6)/N^1/2 = 0.7 and solve for N. Round up to 29.