Margin of error E=0.30 populations standard deviation =2.5. Population means with 95% confidence. What I the required sample size (round up to the whole number)

To calculate the required sample size, we can use the formula:

n = \frac{{Z^2 \cdot \sigma^2}}{{E^2}}

Where:
- \(n\) is the required sample size
- \(Z\) is the z-score corresponding to the desired confidence level (in this case, for 95%, \(Z = 1.96\))
- \(\sigma\) is the population standard deviation
- \(E\) is the margin of error

Plugging in the given values:

n = \frac{{1.96^2 \cdot 2.5^2}}{{0.3^2}}

Simplifying the equation:

n = \frac{{3.8416 \cdot 6.25}}{{0.09}}

n = \frac{{24.01}}{{0.09}}

n \approx 266.778

Rounding up to the nearest whole number, the required sample size is:

Answer: The required sample size is 267.