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Statistics, STD2 S5 2016 HSC 30d

The formula to calculate `z`-scores can be rearranged to give

`mu = x-\sigma z`

 

where    `mu` is the mean
  `x` is the score
  `sigma` is the standard deviation
  `z` is the `z`-score
  1. In an examination, Aaron achieved a score of 88, which corresponds to a `z`-score of 2.4.
  2. Substitute these values into the rearranged formula above to form an equation.   (1 mark)

    --- 1 WORK AREA LINES (style=lined) ---

  3. In the same examination, Brock achieved a score of 52, which corresponds to a `z`-score of  –1.2.
  4. Using this information, form another equation and solve it simultaneously with the equation from part (a) to find the values of `mu` and `\sigma`.   (3 marks)

    --- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

a.    `mu = 88-2.4\sigma`

b.    `64`

Show Worked Solution

a.    `mu = 88-2.4\sigma`
 

b.    `mu = 52 + 1.2\sigma\ …\ (1)`

♦♦ Mean mark (b) 32%.
COMMENT: This is primarily a simultaneous equation problem requiring normal distribution knowledge.

`mu = 88-2.4\sigma \ …\ (2)`
 

`text(Subtract)\ \ (2)-(1):`

`0= 36-3.6\sigma\ \ =>\ \ \sigma= 10`

 
`text(Substitute)\ \ \sigma = 10\ \ text(into)\ (1):`

`mu= 52 + 1.2 xx 10= 64`

Filed Under: DS5/6 - Normal Distribution and Sampling, Normal Distribution, The Normal Distribution (Y12) Tagged With: Band 4, Band 5, smc-6919-60-X-topic, smc-995-10-Single z-score

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