You are a researcher studying the lifespan of a certain species of bacteria. A preliminary sample of 35 bacteria reveals a sample mean of $\bar{x}=72$ hours with a standard deviation of $s=5.2$ 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 $99 \%$ 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.
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Answer
\(\boxed{n = 136}\)
Step 1 ：Given that the margin of error (E) is 0.7, the standard deviation (s) is 5.2, and the Z-score for a 99% level of confidence is approximately 2.576.
Step 2 ：We can use the formula for the sample size (n) which is \(n = (Z * s / E)^2\).
Step 3 ：Substitute the given values into the formula: \(n = (2.576 * 5.2 / 0.7)^2\).
Step 4 ：Calculate the value of n to get approximately 135.6.
Step 5 ：Since we can't have a fraction of a bacterium, we round up to the nearest whole number to get the final sample size.
Step 6 ：\(\boxed{n = 136}\)
