IB Mathematics Applications & Interpretation Question Type 2: Finding the Distribution of the Sample Mean Using CLT Exercises

Let $X_1,X_2,\dots ,X_n$ be i.i.d. random variables with mean $5$ and standard deviation $2$.

Find the smallest value of $n$ such that $P\left(|\bar{X}-5|<0.5\right)\ge 0.95$ using the central limit theorem.

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 $X_1,X_2,\dots ,X_n$ be independent and identically distributed random variables with mean $\mu =500$ and standard deviation $\sigma =50$.

Using the central limit theorem, find the minimum value of $n$ such that $P(490\le \bar{X}\le 510)\ge 0.99$.

Let $X_i$ be independent and identically distributed (i.i.d.) random variables with mean $50$ and standard deviation $10$. For a sample size of $n=64$, use the central limit theorem to approximate $P(\bar{X}>52)$, where $\bar{X}$ is the sample mean.

Let $X_i$ be independent and identically distributed (i.i.d.) random variables with mean $15$ and standard deviation $3$. For a sample of size $n=36$, approximate $P(14<\bar{X}<16)$ using the Central Limit Theorem.

Let $X_1,X_2,X_3,X_4$ 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 $X_1,X_2,\dots ,X_{36}$ be independent and identically distributed random variables with mean $\mu =0$ and standard deviation $\sigma =1$. Let $\bar{X}$ be the sample mean. Use the central limit theorem to approximate $P\left(|\bar{X}|<0.5\right)$.

Let $X_1,X_2,\dots ,X_{100}$ be independent and identically distributed (i.i.d.) random variables with mean $\mu =8$ and standard deviation $\sigma =4$. Use the Central Limit Theorem to find an approximate value for $P(\bar{X}<7.5)$.

Let $X_i$ be i.i.d. random variables with mean $200$ and standard deviation $30$. For $n=81$, approximate $P(\bar{X}<190)$ using the Central Limit Theorem.

Let $X_1,X_2,\dots ,X_{50}$ be independent and identically distributed random variables from a Poisson distribution such that $X_i\sim \text{Po}(3)$.

Approximate $P(\bar{X}\le 3.2)$ using the central limit theorem, where $\bar{X}$ is the sample mean.

Let $X_1,X_2,\dots ,X_n$ be a sequence of independent and identically distributed (i.i.d.) random variables, each having a uniform distribution on the interval $[0,1]$.

For $n=60$, use the central limit theorem to approximate $P(\bar{X}>0.6)$, where $\bar{X}$ is the sample mean.

For $n=25$, find the approximate distribution of the sample mean $\bar{X}$ using the central limit theorem.

Let $X_i$ be i.i.d. random variables with mean $120$ and standard deviation $20$. For $n=25$, use the central limit theorem to approximate $P(110\le \bar{X}\le 130)$.

Let $X_i$ be i.i.d. random variables with mean $10$ and standard deviation $2$. For $n=16$, use the central limit theorem to find an approximation for $P(\bar{X}>11)$.

Let $X_i$ be i.i.d. exponential random variables with mean $2$ and variance $4$. For $n=36$, find the approximate distribution of the sample mean, $\bar{X}$, using the Central Limit Theorem.