You are a researcher studying the lifespan of a certain species of bacteria. A preliminary sample of 25 bacteria reveals a sample mean of 72 hours with a standard deviation of 4 hours. You would like to estimate the mean lifespan for this species of bacteria to within a margin of error of 0.5 hours at a 95% level of confidence.
What sample size should you gather to achieve a 0.5 hour margin of error? Round your answer up to the nearest whole number.
n =
bacteria
To determine the required sample size, we can use the formula for sample size calculation for estimating a population mean:
n = (Z * σ / E)^2
Where:
n = sample size
Z = Z-score (corresponding to the desired level of confidence)
σ = standard deviation of the population (or, in this case, the sample)
E = margin of error
In this case, we are given:
Z = 1.96 (corresponding to a 95% level of confidence, as per the standard normal distribution)
σ = 4 hours
E = 0.5 hours
Plugging in these values into the formula:
n = (1.96 * 4 / 0.5)^2
n = (7.84 / 0.5)^2
n = 15.68^2
n ≈ 246
Therefore, you should gather a sample size of 246 bacteria to achieve a margin of error of 0.5 hours at a 95% level of confidence.