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Question 1

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.

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Question 2

Sampling distributions and 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.

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Question 3

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 $P(490 \le \bar{X} \le 510) \ge 0.99$ .

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Question 4

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.

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Question 5

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.

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Question 6

A random variable $X$ follows a Bernoulli distribution with parameter $p$ . A sample of size $n$ is taken and the sample mean is denoted by $\bar{X}$ .

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 $n=4$ .

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Question 7

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 $P\left(|\bar{X}| < 0.5\right)$ .

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Question 8

The Central Limit Theorem (CLT) states that for a sufficiently large sample size $n$ , the sampling distribution of the sample mean $\bar{X}$ will be approximately normal with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$ , provided the population has a finite mean $\mu$ and standard deviation $\sigma$ .

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 $P(\bar{X} < 7.5)$ .

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Question 9

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.

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Question 10

Topic 4: Statistics and probability — 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.

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Question 11

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.

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Question 12

Let $X_1, X_2, \dots, X_n$ be independent and identically distributed (i.i.d.) random variables with mean $\mu = 20$ and standard deviation $\sigma = 5$ .

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

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Question 13

The central limit theorem states that for a random sample of size $n$ from a population with mean $\mu$ and standard deviation $\sigma$ , the sample mean $\bar{X}$ is approximately normally distributed with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$ , provided $n$ is sufficiently large.

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 $P(110 \le \bar{X} \le 130)$ .

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Question 14

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 $P(\bar{X} > 11)$ .

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Question 15

This question assesses the application of the Central Limit Theorem (CLT) to determine the approximate sampling distribution of the sample mean for a large sample size ( $n=36$ ). Students must identify the population mean and variance for the given distribution and use them to calculate the parameters of the normal approximation.

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.

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