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CIE A Level Maths: Probability & Statistics 2复习笔记1.2.2 Distribution of Sample Means

Category: A-level课程, 教材笔记, 福利干货 Date: 2022年8月18日下午3:02

Sample Mean Distribution

What is the distribution of the sample means?

For any given population it can often be difficult or impractical to find the true value of the population mean, µ
- The population could be too large to collect data using a census or
- Collecting the data could compromise the individual data values and therefore taking a census could destroy the population
- Instead, the population mean can be estimated by taking the mean from a sample from within the population
- It is inversely proportional to the square root of the sample size
- This means that the greater the sample size, the smaller the value of the standard deviation and the narrower the distribution of the sample means

5-3-1-sample-means-diagram-1

Worked Example

5-3-1-sample-means-we-1

Exam Tip

Look carefully at the distribution given to determine whether the variance or the standard deviation has been given.

Central Limit Theorem

Does the distribution of the sample means always follow a normal distribution?

What is the central limit theorem?

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.

Find the probability that the mean of Susie’s 40 numbers is less than 13. Explain whether it was necessary to use the Central Limit Theorem in your calculation.

1-2-2-central-limit-theorem-we-solution

Exam Tip

If asked to explain whether it was necessary to use the Central Limit Theorem, check whether the population follows a normal distribution, if it does not then check the size of the sample. If your answer is yes comment on both of these things.

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