Sample Mean Distribution (DP IB Maths: AI HL)

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Dan

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Dan

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Combinations of Normal Variables

What is a linear combination of normal random variables?

  • Suppose you have n independent normal random variables X subscript i tilde straight N invisible function application open parentheses mu subscript i comma blank sigma subscript i superscript 2 close parentheses for i = 1,2,3, ..., n
  • A linear combination is of the form X equals a subscript 1 X subscript 1 plus a subscript 2 X subscript 2 plus blank horizontal ellipsis plus a subscript n X subscript n plus b where ai and b are constants
  • The mean and variance can be calculated using results from random variables
    • straight E invisible function application open parentheses X close parentheses equals a subscript 1 mu subscript 1 plus a subscript 2 mu subscript 2 plus blank horizontal ellipsis plus a subscript n mu subscript n plus b
    • Var invisible function application open parentheses X close parentheses equals a subscript 1 superscript 2 sigma subscript 1 superscript 2 plus a subscript 2 superscript 2 sigma subscript 2 superscript 2 plus blank horizontal ellipsis plus a subscript n superscript 2 sigma subscript n superscript 2
      • The variables need to be independent for this result to be true
  • A linear combination of n independent normal random variables is also a normal random variable itself
    •  X tilde straight N invisible function application open parentheses a subscript 1 mu subscript 1 plus a subscript 2 mu subscript 2 plus blank horizontal ellipsis plus a subscript n mu subscript n plus b comma space a subscript 1 superscript 2 sigma subscript 1 superscript 2 plus a subscript 2 superscript 2 sigma subscript 2 superscript 2 plus blank horizontal ellipsis plus a subscript n superscript 2 sigma subscript n superscript 2 close parentheses
    • 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
    • top enclose X equals fraction numerator X subscript 1 plus X subscript 2 plus blank horizontal ellipsis plus X subscript n over denominator n end fraction
  • For an individual sample the sample mean x with bar on top can be calculated
    • This is also called a point estimate
    • top enclose X is the distribution of the point estimates

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
  • straight E invisible function application open parentheses X with bar on top close parentheses equals straight E invisible function application open parentheses fraction numerator X subscript 1 plus X subscript 2 plus blank horizontal ellipsis plus X subscript n over denominator n end fraction close parentheses equals fraction numerator straight E invisible function application open parentheses X subscript 1 close parentheses plus straight E invisible function application open parentheses X subscript 2 close parentheses plus blank horizontal ellipsis plus straight E left parenthesis X subscript n right parenthesis over denominator n end fraction equals fraction numerator mu plus mu plus blank horizontal ellipsis plus mu over denominator n end fraction equals fraction numerator n mu over denominator n end fraction equals mu
  • Var invisible function application open parentheses X with bar on top close parentheses equals Var invisible function application open parentheses fraction numerator X subscript 1 plus X subscript 2 plus blank horizontal ellipsis plus X subscript n over denominator n end fraction close parentheses equals fraction numerator Var invisible function application open parentheses X subscript 1 close parentheses plus Var invisible function application open parentheses X subscript 2 close parentheses plus blank horizontal ellipsis plus Var left parenthesis X subscript n right parenthesis over denominator n ² end fraction equals fraction numerator sigma ² plus sigma ² plus blank horizontal ellipsis plus sigma ² over denominator n ² end fraction equals fraction numerator n sigma ² over denominator n ² end fraction equals sigma squared over n
  • Therefore you divide the variance of the population by the size of the sample to get the variance of the sample mean distribution
    • X tilde straight N invisible function application open parentheses mu comma sigma squared close parentheses rightwards double arrow X with bar on top tilde straight N invisible function application open parentheses mu comma sigma squared over n close parentheses

Worked example

Amber makes a cup of tea using a hot drink vending machine. When the hot water button is pressed the machine dispenses  Wml of hot water and when the milk button is pressed the machine dispenses M ml of milk. It is known that W blank tilde straight N invisible function application open parentheses 100 comma blank 15 squared close parentheses and M blank tilde straight N invisible function application open parentheses 10 comma blank 2 squared close parentheses

To make a cup of tea Amber presses the hot water button three times and the milk button twice. The total amount of liquid in Amber’s cup is modelled by C ml.

a)
Write down the distribution of C.

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

c)
Amber makes 15 cups of tea and calculates the mean C with bar on top. Write down the distribution of C with bar on top.

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

Central Limit Theorem

What is the Central Limit Theorem?

  • The Central Limit Theorem says that if a sufficiently large random sample is taken from any distribution X then the sample mean distribution X with bar on top can be approximated by a normal distribution
    • In your exam n > 30 will be considered sufficiently large for the sample size
  • Therefore X with bar on top can be modelled by straight N invisible function application open parentheses mu comma sigma squared over n close parentheses
    • 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
  • If X is normally distributed then X with bar on top is normally distributed
    • This is true regardless of the size of the sample
    • The Central Limit Theorem is not needed
  • If X is not normally distributed then X with bar on top is approximately normally distributed
    • Provided the sample size is large enough
    • The Central Limit Theorem is needed

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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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.