1. The problem is to represent the approximation of binomial distributions to normal distributions for probabilities $p=0.5$ and $p=0.05$ with sample sizes $n=3, 25, 50$. 2. The binomial distribution $B(n,p)$ can be approximated by a normal distribution $N(\mu, \sigma)$ where: $$\mu = np$$ $$\sigma = \sqrt{np(1-p)}$$ This approximation is better for larger $n$ and when $p$ is not too close to 0 or 1. 3. For each $p$ and $n$, calculate: - Mean: $\mu = np$ - Standard deviation: $\sigma = \sqrt{np(1-p)}$ 4. Plot the binomial probability mass function (pmf) and overlay the normal probability density function (pdf) with continuity correction by shifting $x$ by 0.5. 5. Example calculations for $p=0.5$: - $n=3$: $\mu=1.5$, $\sigma=\sqrt{3\times0.5\times0.5}=0.866$ - $n=25$: $\mu=12.5$, $\sigma=\sqrt{25\times0.5\times0.5}=2.5$ - $n=50$: $\mu=25$, $\sigma=\sqrt{50\times0.5\times0.5}=3.535$ 6. Example calculations for $p=0.05$: - $n=3$: $\mu=0.15$, $\sigma=\sqrt{3\times0.05\times0.95}=0.378$ - $n=25$: $\mu=1.25$, $\sigma=\sqrt{25\times0.05\times0.95}=1.089$ - $n=50$: $\mu=2.5$, $\sigma=\sqrt{50\times0.05\times0.95}=1.54$ 7. The plots show that as $n$ increases, the binomial distribution approaches the normal distribution more closely. Final answer: The normal approximation to the binomial distribution improves with larger $n$ and when $p$ is closer to 0.5, as shown by the calculated means and standard deviations and the graphical overlays.