natalia estimates that she wins a board game 72% of the time. how large of a random sample is required to…

Explanation:

Step1: Identify the formula

The formula for sample - size $n$ in proportion is $n=\frac{z^{2}\cdot p(1 - p)}{E^{2}}$, where $z$ is the z - score, $p$ is the estimated proportion, and $E$ is the margin of error. For a 95% confidence interval, the z - score $z = 1.96$ (from the standard normal distribution table). The estimated proportion $p=0.72$, and the margin of error $E = 0.06$.

Step2: Calculate $p(1 - p)$

$p(1 - p)=0.72\times(1 - 0.72)=0.72\times0.28 = 0.2016$.

Step3: Calculate $n$

$n=\frac{(1.96)^{2}\times0.2016}{(0.06)^{2}}$. First, calculate $(1.96)^{2}=3.8416$ and $(0.06)^{2}=0.0036$. Then, $n=\frac{3.8416\times0.2016}{0.0036}=\frac{0.77446656}{0.0036}\approx215.13$. Since the sample size $n$ must be an integer, we round up to $n = 216$.

Answer:

216