# Using Chebyshev and other inequality formulas (maybe even Central Limit Theorem)

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In summary, Turner's syndrome is a rare chromosomal disorder affecting about 1 in 2000 girls in the United States. 1 in 10 girls with Turner's syndrome also have an abnormal narrowing of the aorta. To calculate the probability of a certain number of girls being affected, the binomial distribution can be used. For question (b.), a normal approximation can be used instead. Additionally, for question (a.), a Poisson distribution is also appropriate due to the low probability of the disorder.
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Turner's syndrome is a rare chromosomal disorder in which girls have only one X chromosome. It affects about 1 in 2000 girls in the United States. About 1 in 10 girls with Turner's syndrome also suffer from an abnormal narrowing of the aorta.

a. In a group of 4000 girls, what is the probability that there are 0,1,2, or at least 3 girls affected with Turner's syndrome?

b. In a group of 170 girls affected with Turner's syndrome, what is the probability that at least 20 of them suffer from an abnormal narrowing of the aorta?

For both questions the most direct approach would be using the binomial distribution, although for (b.) a normal approximation would make calculation easier.

For a., a Poisson distribution is also appropriate (and probably easier to work with than the Binomial), since the probability of the disorder is so low.

## What is Chebyshev's inequality and how is it used?

Chebyshev's inequality is a mathematical formula used to approximate the probability that a random variable will deviate from its mean by a certain amount. It states that for any random variable, the probability of it falling within k standard deviations of the mean is at least 1 - 1/k^2. This formula is often used to determine the likelihood of extreme or rare events occurring.

## How does Chebyshev's inequality compare to other inequality formulas?

Chebyshev's inequality is a more general formula than other inequality formulas, such as Markov's inequality and Chernoff's bound. It provides a tighter bound on the probability of deviation from the mean, but is more complex to calculate. Markov's inequality, for example, only applies to non-negative random variables, while Chebyshev's inequality can be applied to any random variable with a finite variance.

## What is the Central Limit Theorem and how is it used in statistics?

The Central Limit Theorem states that as the sample size increases, the distribution of sample means from any population will approach a normal distribution. This theorem is important in statistics because it allows us to make inferences about a population based on a smaller sample. It is also used to calculate confidence intervals and hypothesis testing.

## How can the Central Limit Theorem be applied in real-world situations?

The Central Limit Theorem has many practical applications, such as in market research where a sample of consumers is used to make predictions about a larger population. It is also used in quality control, to determine if a production process is meeting specifications. In finance, it can be used to analyze investment returns and in medical research, it can help determine the effectiveness of a treatment.

## What are the limitations of using Chebyshev and other inequality formulas?

While Chebyshev and other inequality formulas can be useful tools in probability and statistics, they have their limitations. These formulas assume that the data is independent and identically distributed, and that the sample size is large enough for the Central Limit Theorem to apply. If these assumptions are not met, the results may not be accurate. Additionally, these formulas provide approximations and not exact probabilities, so they should be used with caution and in conjunction with other statistical methods.

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