N2018 P2 Q10

N2018 P2 Q10

Junior College 2
15 marks
Free

In this question you should state the parameters of any distributions that you use.

A manufacturer produces specialist light bulbs. The masses in grams of one type of light bulb have the normal distribution \(N(50,1.5^2)\).

  1. Sketch the distribution for masses between 40 grams and 60 grams.[2]
  2. Find the probability that the mass of a randomly chosen bulb is less than 50.4 grams.[1]

Each light bulb is packed into a randomly chosen box. The masses of the empty boxes have the distribution \(N(75,2^2)\).

  1. Find the probability that the total mass of 4 randomly chosen empty boxes is more than 297 grams.[2]
  2. Find the probability that the total mass of a randomly chosen light bulb and a randomly chosen box is between 124.9 and 125.7 grams.[3]

In order to protect the bulbs in transit each bulb is surrounded by padding before being packed in a box. The mass of the padding is modelled as 30% of the mass of the bulb.

  1. The probability that the total mass of a box containing a bulb and padding is more than \(k\) grams is \(0.9\). Find \(k\).[4]
  2. Find the probability that the total mass of 4 randomly chosen boxes, each containing a bulb and padding, is more than 565 grams.[3]

Default solution

Let \(X\sim N(50,1.5^2)\) be bulb mass and \(B\sim N(75,2^2)\) be independent box mass, in grams.

(i) Sketch a symmetric normal density on \(40\le x\le60\), centred at \(50\), with standard deviation \(1.5\) and tails approaching the axis.

(ii) \(P(X<50.4)=\Phi\!\left(\frac{50.4-50}{1.5}\right)=0.605\).

(iii) \(\sum_{j=1}^4B_j\sim N(300,16)\); \(P\!\left(\sum B_j>297\right)=1-\Phi(-0.75)=0.773\).

(iv) \(X+B\sim N(125,6.25)\); \(P(124.9<X+B<125.7)=\Phi(0.28)-\Phi(-0.04)=0.126\).

(v) \(Y=1.3X+B\sim N(140,7.8025)\). \(P(Y>k)=0.9\Rightarrow k=140+\Phi^{-1}(0.1)\sqrt{7.8025}=136\text{ g}\) (3 s.f.).

(vi) \(\sum_{j=1}^4Y_j\sim N(560,31.21)\); \(P\!\left(\sum Y_j>565\right)=1-\Phi\!\left(\frac5{\sqrt{31.21}}\right)=0.185\).

Answer:(ii) \(0.605\); (iii) \(0.773\); (iv) \(0.126\); (v) \(136\text{ g}\); (vi) \(0.185\)

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