Question

What is the Central Limit Theorem? Discuss an example of its application.

Answer 1

Answer:

From given data,

What is the Central Limit Theorem? Discuss an example of its application.

The Central Limit Theorem :

The central limit theorem states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough.

How large is "large enough" The answer depends on two factors.

• Requirements for accuracy. The more closely the sampling distribution needs to resemble a normal distribution, the more sample points will be required.
• The shape of the underlying population. The more closely the original population resembles a normal distribution, the fewer sample points will be required.

KEY TAKEAWAYS

• The central limit theorem (CLT) states that the distribution of sample means approximates a normal distribution as the sample size gets larger.
• Sample sizes equal to or greater than 30 are considered sufficient for the CLT to hold.
• A key aspect of CLT is that the average of the sample means and standard deviations will equal the population mean and standard deviation.
• A sufficiently large sample size can predict the characteristics of a population accurately.

An example of its application

It is believed that nearsightedness affects about 8% of all children. 194 incoming children have their eyesight tested. Can the CLT be used in this situation?

● Randomization: We have to assume there isn't some factor in the region that makes it more likely these kids have vision problems.

● 10% Rule: The population is “all children” - this is in the millions. 194 is less than 10% of the population.

● np=194*.08=15.52, nq=194*.92=176.48 We have to make one assumption when using the CLT in this situation.

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