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 $P(|\bar{X} - 100| < 2) \ge 0.95$.