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Theory

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Procedure

Experiments

Posttest

References

Contributors

Feedback

Central Limit Theorem

Choose difficulty:

Beginner

Intermediate

Advanced

1: How does the characteristic function help prove the CLT?

a: It simplifies the calculation of the population mean. Explanation

Explanation

b: It shows that the distribution of normalized sums converges to

\exp\left(\frac{-t^2}{2}\right)

Explanation

c: It demonstrates the uniqueness of the sample mean. Explanation

Explanation

d: It avoids the use of independence in CLT proofs. Explanation

Explanation

2: What happens to the rate of convergence to normality in the CLT if the population distribution is highly skewed?

a: Convergence happens faster. Explanation

Explanation

b: Convergence happens slower. Explanation

Explanation

c: Convergence does not occur. Explanation

Explanation

d: Convergence depends on the sample size only, not skewness. Explanation

Explanation

3: Consider a distribution where

X_i

has infinite variance. Can the CLT still apply?

a: Yes, as long as the mean is finite. Explanation

Explanation

b: Yes, but only if

X_i

are not identically distributed. Explanation

Explanation

c: No, because variance is crucial for the CLT to hold. Explanation

Explanation

d: No, because the CLT does not depend on variance. Explanation

Explanation

4: Which statement about the sample variance represented by

\hat{\sigma^2} = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2

, where the sample mean is defined as \( \bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i \); is true in the context of CLT?

\hat{\sigma^2}

is unbiased for

\sigma^2

. Explanation

Explanation

\hat{\sigma^2}

converges to

\frac{\sigma^2}{n}

n \to \infty

. Explanation

Explanation

\hat{\sigma^2}

converges to a normal distribution as

n \to \infty

. Explanation

Explanation

\hat{\sigma^2}

is irrelevant to the CLT. Explanation

Explanation

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