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

To estimate the mean lifespan of bacteria with a margin of error of 0.7 hours at a 90% confidence level, a sample size of 117 bacteria is needed.

Step-by-step explanation:

Sample Size Calculation for Estimating a Mean

To determine the sample size needed to estimate the mean lifespan of a species of bacteria with a given margin of error and confidence level, the following formula derived from the Central Limit Theorem is used:

n = (Z* × s / E)^2

Where:

• n = required sample size
• Z* = Z-score corresponding to the desired confidence level
• s = sample standard deviation
• E = margin of error

Here, a 90% confidence level corresponds to a Z-score (Z*) of approximately 1.645 (from Z-table). The provided sample standard deviation (s) is 4.6 hours, and the desired margin of error (E) is 0.7 hours. Plugging these values into the formula gives us:

n = (1.645 × 4.6 / 0.7)^2

Calculating this, we get:

n = (1.645 × 6.5714)^2 ≈ (10.8158)^2 ≈ 116.9817

Since we cannot have a fraction of a bacterium, we round up to the nearest whole number:

n = 117 bacteria