4.15 Central limit theoremIB Maths: Applications and Interpretation HL: Flashcards

What these 13 flashcards ask

A linear combination of independent normal variables is...?
For independent X,Y, find Var(X-Y).
X\sim N(\mu,\sigma^2): distribution of \bar{X} for a sample of n?
Standard deviation of \bar{X} (the standard error)?
Distribution of the total of n independent copies of X\sim N(\mu,\sigma^2)?
State the central limit theorem.
What sample size is considered large enough in IB examinations?
When is \bar{X} exactly normal?
What happens to the spread of \bar{X} as n increases?
Variance of 2X compared with X1+X2 (independent copies of X)?
Does the CLT help for a single observation from a skewed population?
How can you visualise the CLT?
Why must a sample size found from an inequality be rounded up?

Exam questions on 4.15 Central limit theorem

The mass $X$ kg of a bag of flour is normally distributed with mean $1.00$ kg and standard deviation $0.04$ kg. A random sample of $9$ bags is taken, with masses independent of each other, and $\bar{X}$ is the mean mass of the sample.
Explain why $\bar{X}$ is exactly normally distributed here, even though the sample size is only $9$ .2 marks
The mass of an adult passenger, $X$ kg, is normally distributed with mean $78$ kg and standard deviation $11$ kg. Passenger masses are independent of each other.
A lift is overloaded if the total mass of six passengers exceeds $500$ kg. Use your GDC to find the probability that the lift is overloaded.2 marks
The waiting time $T$ minutes of a caller at a call centre has mean $6.0$ minutes and standard deviation $4.5$ minutes. The distribution of $T$ is positively skewed, so it is not normal. A random sample of $50$ calls is taken and $\bar{T}$ is the mean waiting time of the sample.
State the approximate distribution of $\bar{T}$ , giving the reason why the approximation is valid and its parameters.3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).