IB DP Maths: AI HL复习笔记4.9.1 Sample Mean Distribution
Combinations of Normal Variables
What is a linear combination of normal random variables?
This can be used to find probabilities when combining normal random variables
What is meant by the sample mean distribution?
- Suppose you have a population with distribution X and you take a random sample with n observations X1, X2, ..., Xn
- The sample mean distribution is the distribution of the values of the sample mean
What does the sample mean distribution look like when X is normally distributed?
- If the population is normally distributed then the sample mean distribution is also normally distributed
Worked Example
b)Find the probability that the total amount of liquid in Amber's cup exceeds 360 ml.
Central Limit Theorem
What is the Central Limit Theorem?
μ is the mean of X
-
- σ² is the variance of X
- n is the size of the sample
Do I always need to use the Central Limit Theorem when working with the sample mean distribution?
- No – the Central Limit Theorem is not needed when the population is normally distributed
Worked Example
The integers 1 to 29 are written on counters and placed in a bag. The expected value when one is picked at random is 15 and the variance is 70. Susie randomly picks 40 integers, returning the counter after each selection.
a)Find the probability that the mean of Susie's 40 numbers is less than 13.
b)Explain whether it was necessary to use the Central Limit Theorem in your calculation.
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