IB Mathematics Applications & Interpretation Question Type 3: Solving Probability Questions of Other Distributions Using CLT Exercises

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

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 $0.35$.

[4 marks]

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 $\bar{X}$ is below $7.5$?

Let $X_i\sim \text{Exp}(\lambda =1/2)$, so that the population mean $\mu =2$ and standard deviation $\sigma =2$.

Calculate an approximation for $P(1.8<\overline{X}<2.2)$, where $\overline{X}$ is the sample mean.

Suppose $X$ has mean $5$ and standard deviation $3$. Determine the minimum $n$ such that $P(\overline{X}>6)\le 0.01$ by using the Central Limit Theorem.

Let $X_1,X_2,\dots ,X_{36}$ be a random sample of size $n=36$ from an exponential distribution with parameter $\lambda =1$, such that the population mean $\mu =1$ and population standard deviation $\sigma =1$.

Let $\overline{X}$ denote the sample mean. Use the central limit theorem to find an approximate value for $P(\overline{X}<0.8)$.

Independent samples from two populations yield sample mean $\overline{X}$ with mean $20$, standard deviation $3$, $n_X=36$, and sample mean $\overline{Y}$ with mean $18$, standard deviation $4$, $n_Y=49$. Calculate an approximate value for $P(\overline{X}-\overline{Y}>3)$.

Let $X_1,X_2,\dots ,X_{100}$ be independent random variables such that $X_i\sim \text{Poisson}(20)$ for each $i$.

Approximate the probability that the sum $S={\sum }_{i=1}^{100}{X}_i$ exceeds $2100$.

Let $X_1,X_2,\dots ,X_{30}$ be a sequence of independent random variables such that $X_i\sim \text{Po}(10)$ for each $i$.

Approximate the probability that the sample mean, $\bar{X}$, exceeds $12$.

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

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

A process yields measurements with mean $100$ and standard deviation $15$. Find the minimum sample size $n$ such that $P(|\bar{X}-100|<2)\ge 0.95$.