When calculating the sample size for a confidence interval, the margin of error formula \( E = Z \cdot \frac{\sigma}{\sqrt{n}} \) is very useful. The key step is isolating \( n \) after finding the margin of error. Be sure to square your result for \( \sqrt{n} \) to get the sample size. Also, remember to round the sample size to the nearest whole number since you cannot have a fraction of a sample.
The formula for the margin of error (E) in a confidence interval is given by \(E = Z \cdot \frac{\sigma}{\sqrt{n}}\).
In this formula, \(E\) represents half the width of the interval, \(Z\) is 1.96, and \(\sigma\) is 25.
The calculated width of the confidence interval is \(170 - 160 = 10\). Therefore, the margin of error is \(E = \frac{10}{2} = 5\).
Substituting these values into the margin of error formula yields \(5 = 1.96 \cdot \frac{25}{\sqrt{n}}\).
Solving for \(n\), we first find \(\sqrt{n} = \frac{1.96 \cdot 25}{5} = 9.8\).
Then, squaring both sides gives \(n = 9.8^2 = 96.04\).
Consequently, the required sample size is \(n = 96\).
| \(\text{Length (in mm)}\) | 70-80 | 80-90 | 90-100 | 100-110 | 110-120 | 120-130 | 130-140 |
|---|---|---|---|---|---|---|---|
| \(\text{Number of leaves}\) | 3 | 5 | 9 | 12 | 5 | 4 | 2 |