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IB DP Maths: AI HL復習筆記4.9.1 Sample Mean Distribution

Category: IB課程, 教材筆記, 福利干貨 Date: 2022年7月21日 下午5:12

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 distributedWorked Example

4-9-1-ib-ai-hl-linear-normal-comb-a-we-solution

b)Find the probability that the total amount of liquid in Amber's cup exceeds 360 ml.

4-9-1-ib-ai-hl-linear-normal-comb-b-we-solution

4-9-1-ib-ai-hl-linear-normal-comb-c-we-solution

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.

4-9-1-ib-ai-hl-central-limit-theorem-a-we-solution

b)Explain whether it was necessary to use the Central Limit Theorem in your calculation.

4-9-1-ib-ai-hl-central-limit-theorem-b-we-solution

 

 

 

 

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