The shopkeeper places \(100\) randomly chosen potatoes on a weighing machine. Find the probability that their total weight exceeds \(10\) kilograms.

Find the minimum number of randomly selected potatoes which are needed to ensure that their total weight exceeds \(10\) kilograms with probability greater than \(0.95\).

let \(T\) denote the total weight, then

\(T \sim N(9800,25600)\)    
(M1)(A1)

\({\rm{P}}(T > 10000) = 0.106\)     A1

[3 marks]

let there be \(n\) potatoes, in this case,

\(T \sim {\rm{N}}(98n,256n)\)     A1

we require

\({\rm{P}}(T > 10000) > 0.95\)     (M1)

or equivalently

\({\rm{P}}(T \le 10000) < 0.05\)     A1

standardizing,

\({\rm{P}}\left( {Z \le \frac{{10000 - 98n}}{{16\sqrt n }}} \right) < 0.05\)     A1

\(\frac{{10000 - 98n}}{{16\sqrt n }} < - 1.6449 \ldots \)     (A1)

\(98n - 26.32\sqrt n  - 10000 > 0\)     A1

solving the corresponding equation, \(n = 104.7 \ldots \)     (A1)

the required minimum value is \(105\)     A1

Note: Part (b) could also be solved using SOLVER and normalcdf, or by trial and improvement.

Note: Allow the use of \( = \) instead of \( < \) and \( > \) throughout.

[8 marks]