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Worked solution

Find three probabilities for package mass

Packages from a packing machine have a mass which is normally distributed with mean 200 g, and standard deviation 2 g. Find the probability that a package from the machine weighs

(i) between 198.5 and 199.5 g,

(ii) less than 197 g,

(iii) more than 200.5 g.

CJC · Discussion · Question 1

Answer

(i) 0.175; (ii) 0.0668; (iii) 0.401 (3 s.f.).

Define the model

Let X be mass in grams. Then X∼N(200,22)X\sim N(200,2^2); standardise using Z=(X−200)/2Z=(X-200)/2.

How did this part go?

(i) Between the two bounds

P(198.5<X<199.5)=P(−0.75<Z<−0.25)=Φ(−0.25)−Φ(−0.75)≈0.175P(198.5<X<199.5)=P(-0.75<Z<-0.25)=\Phi(-0.25)-\Phi(-0.75)\approx0.175 to 3 s.f.

How did this part go?

(ii) Lower tail

P(X<197)=P(Z<−1.5)≈0.0668P(X<197)=P(Z<-1.5)\approx0.0668 to 3 s.f.

How did this part go?

(iii) Upper tail

P(X>200.5)=P(Z>0.25)=1−Φ(0.25)≈0.401P(X>200.5)=P(Z>0.25)=1-\Phi(0.25)\approx0.401 to 3 s.f.

How did this part go?

Common mistakes

A probability beyond an upper bound uses the upper tail, not the cumulative lower tail.

Solution notes

All source values and rounding retained; standardised bounds are added to explain the calculator results.

Read the hint