Question Type 2: Finding the distribution of the sample mean using CLT Exercises

Let $X_i$ be independent Bernoulli random variables with $p=0.5$. Using the central limit theorem, find the approximate distribution of the sample mean $\bar X$ for $n=4$.

Let $X_i$ be i.i.d. with mean $20$ and standard deviation $5$. For $n=25$, find the approximate distribution of $\bar X$ using the central limit theorem.

Let $X_i$ be i.i.d. exponential random variables with mean $2$ (variance $4$). For $n=36$, find the approximate distribution of $\bar X$ using the central limit theorem.

Let $X_i$ be i.i.d. with mean $8$ and standard deviation $4$. For $n=100$, use the central limit theorem to approximate $P(\bar X < 7.5)$.

Let $X_i$ be i.i.d. with mean $15$ and standard deviation $3$. For $n=36$, approximate $P\bigl(14<\bar X<16\bigr)$ using the central limit theorem.

Let $X_i$ be i.i.d. Poisson$(3)$. For $n=50$, approximate $P(\bar X\le3.2)$ using the central limit theorem.

Let $X_i$ be i.i.d. with mean $10$ and standard deviation $2$. For $n=16$, use the central limit theorem to approximate $P(\bar X>11)$.

Let $X_i$ be i.i.d. uniform on $[0,1]$. For $n=60$, use the central limit theorem to approximate $P(\bar X>0.6)$.

Let $X_i$ be i.i.d. with mean $0$ and standard deviation $1$. For $n=36$, use the central limit theorem to approximate $P\bigl(|\bar X|<0.5\bigr)$.

Let $X_i$ be i.i.d. with mean $50$ and standard deviation $10$. For $n=64$, use the central limit theorem to approximate $P(\bar X>52)$.

Let $X_i$ be i.i.d. with mean $100$ and standard deviation $15$. Determine the minimum sample size $n$ so that the standard deviation of $\bar X$ is at most $2$ using the central limit theorem.

Let $X_i$ be i.i.d. with mean $500$ and standard deviation $50$. Find the minimum $n$ such that $P(490\le\bar X\le510)=0.99$ using the central limit theorem.

Let $X_i$ be i.i.d. with mean $200$ and standard deviation $30$. For $n=81$, approximate $P(\bar X<190)$ using the central limit theorem.

Let $X_i$ be i.i.d. with mean $5$ and standard deviation $2$. Find the smallest $n$ such that $P\bigl(|\bar X-5|<0.5\bigr)\ge0.95$ using the central limit theorem.

Let $X_i$ be i.i.d. with mean $120$ and standard deviation $20$. For $n=25$, use the central limit theorem to approximate $P(110\le\bar X\le130)$.