Find k given probability statement

Given a normal distribution with known mean and standard deviation, find the unknown value k from a probability statement like P(X < k) = p or P(X > k) = p.

5 questions · Standard +0.0

2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation
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CAIE S1 2016 November Q1
3 marks Moderate -0.8
1 The random variable \(X\) is such that \(X \sim \mathrm {~N} ( 20,49 )\). Given that \(\mathrm { P } ( X > k ) = 0.25\), find the value of \(k\).
Edexcel S1 2015 January Q7
11 marks Standard +0.3
  1. The birth weights, \(W\) grams, of a particular breed of kitten are assumed to be normally distributed with mean 99 g and standard deviation 3.6 g
    1. Find \(\mathrm { P } ( W > 92 )\)
    2. Find, to one decimal place, the value of \(k\) such that \(\mathrm { P } ( W < k ) = 3 \mathrm { P } ( W > k )\)
    3. Write down the name given to the value of \(k\).
    For a different breed of kitten, the birth weights are assumed to be normally distributed with mean 120 g Given that the 20th percentile for this breed of kitten is 116 g
  2. find the standard deviation of the birth weight of this breed of kitten.
OCR S2 2011 June Q2
7 marks Standard +0.3
2 The random variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that $$\mathrm { P } ( Y < 48.0 ) = \mathrm { P } ( Y > 57.0 ) = 0.0668 .$$ Find the value \(y _ { 0 }\) such that \(\mathrm { P } \left( Y > y _ { 0 } \right) = 0.05\).
Edexcel S1 2003 June Q2
6 marks Moderate -0.5
2. The lifetimes of batteries used for a computer game have a mean of 12 hours and a standard deviation of 3 hours. Battery lifetimes may be assumed to be normally distributed. Find the lifetime, \(t\) hours, of a battery such that 1 battery in 5 will have a lifetime longer than \(t\).
Pre-U Pre-U 9794/1 2011 June Q15
12 marks Standard +0.8
A firm produces chocolate bars whose weights are normally distributed with mean 120 g and standard deviation 6 g.
  1. Bars which weigh more than 114 g are sold at a profit of 15p per bar. The remaining bars are sold at no profit. Show that the expected profit per 100 bars is £12.62. [5]
  2. It is subsequently decided that bars which weigh more than \(x\) g should be sold at a profit of 20p per bar. Those which weigh \(x\) g or less are sold to employees at a profit of 3p per bar. The expected profit per 100 bars is £19.17. Find the value of \(x\). [7]