Question Type 3: Solving probability questions of other distributions using CLT Exercises

Suppose $X$ has mean $50$ and standard deviation $5$. For a sample of size $n=64$, approximate the probability that the sample mean exceeds $52$.

Let $X$ be a random variable with mean $8$ and standard deviation $4$. If $n=100$ independent observations are taken, what is the approximate probability that the sample mean $\overline X$ is below $7.5$?

Let $X_i\sim\text{Poisson}(\lambda=10)$ be independent. For $n=30$, approximate the probability that the sample mean exceeds $12$.

Let $X\sim\text{Binomial}(100,0.2)$. Use a normal approximation to estimate $P(X>30)$.

Let $X_i\sim\text{Exp}(1)$ so $\mu=1$, $\sigma=1$. For $n=36$, approximate $P(\overline X<0.8)$.

Let $X_i\sim\text{Exp}(\lambda=1/2)$ so $\mu=2$, $\sigma=2$. For $n=100$, approximate $P(1.8<\overline X<2.2)$.

A sample of size $n=200$ is drawn from a Bernoulli distribution with $p=0.3$. Approximate the probability that the sample proportion $\hat p$ exceeds $0.35$.

A process yields measurements with mean $100$ and standard deviation $15$. Find the minimum sample size $n$ so that $P(|\overline X-100|<2)\ge0.95$.

Let $X_i\sim U(0,10)$ so $\mu=5$, $\sigma^2=100/12$. For $n=50$, approximate the probability that $\overline X<6$.

Independent samples from two populations yield $\overline X$ with mean $20$, sd $3$, $n_X=36$, and $\overline Y$ with mean $18$, sd $4$, $n_Y=49$. Approximate $P(\overline X-\overline Y>3)$.

Let $X_i\sim\text{Poisson}(20)$. For $n=100$, approximate the probability that the total $S=\sum_{i=1}^{100}X_i$ exceeds $2100$.

Suppose $X$ has mean $5$ and standard deviation $3$. Determine the minimum $n$ such that $P(\overline X>6)\le0.01$ by using the Central Limit Theorem.