Solving P(11<M<12) Using CLT and Standard Normal Distribution CDF

In summary, the problem involves finding the probability of a certain range of values for a variable M, which is calculated using a sample of 100 values from a distribution X. The question can be solved using the central limit theorem, even though the independence of the X values is not explicitly stated. The normal distribution should be used to find the desired probability.
  • #1
kingwinner
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Homework Statement


Let M=(X1+X2+...+X100)/100 where each Xi's has the same distribution of X. Find P(11<M<12) in terms of the cumulative distribution function for the standard normal distribution.


Homework Equations


The Attempt at a Solution


This looks like a "central limit theorem(CLT)" question to me, but with a careful look at the assumptions of CLT, it says that the Xi's must be independent and identically distributed while in this question, there is nothing that indicates independence. How can I solve this problem then?



Thank you for helping!
 
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  • #2
Any stat guy here? Please help...
 
  • #3
Well, the problem says that each X_i has "the same distribution", and I would say that implies that this distribution must be independent of the other X_i's. But we don't even need to worry about this, because the problem goes on to tell us to use the normal distribution, which is what the CLT would tell us to do.
 

What is the Central Limit Theorem?

The Central Limit Theorem states that when independent random variables are added, their sum tends to be normally distributed.

Why is the Central Limit Theorem important?

The Central Limit Theorem is important because it allows us to make inferences about a population based on a sample, even when the population is not normally distributed. This makes it a fundamental tool in statistical analysis.

What are the assumptions of the Central Limit Theorem?

The Central Limit Theorem assumes that the random variables are independent and identically distributed, and that the sample size is large enough (usually considered to be at least 30).

How is the Central Limit Theorem used in practical applications?

The Central Limit Theorem is used to estimate unknown population parameters based on a sample, such as calculating the mean or standard deviation of a population. It is also used in hypothesis testing and constructing confidence intervals.

Does the Central Limit Theorem always hold true?

The Central Limit Theorem is a theoretical concept and may not always hold true in practice. It is most accurate when the underlying population is normally distributed, but in some cases, even with a large sample size, the distribution of the sample may not be normal. Additionally, the Central Limit Theorem is based on assumptions that may not always be met in real-world situations.

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