Normal probabilities with the calculator Problem set A Worked solutions

The lifetime of a type of LED bulb is normally distributed with mean 12 000 hours and standard deviation 1 500 hours. Find the probability that a bulb lasts longer than 14 000 hours.

• \(X \sim N(12 000, 1 500^2)\)
• From the calculator: normal cdf with lower = 14 000, upper = 10^99, μ = 12 000, σ = 1 500
• P = 0.0912

The mass of a ripe melon is normally distributed with mean 1.8 kg and standard deviation 0.25 kg. Find the probability that a melon has mass less than 1.5 kg.

• \(X \sim N(1.8, 0.25^2)\)
• From the calculator: normal cdf with lower = −10^99, upper = 1.5, μ = 1.8, σ = 0.25
• P = 0.115

The resting pulse of adults in a town is normally distributed with mean 68 beats per minute and standard deviation 7. Find the probability that an adult has a pulse between 60 and 75 beats per minute.

• \(X \sim N(68, 7^2)\)
• From the calculator: normal cdf with lower = 60, upper = 75, μ = 68, σ = 7
• P = 0.715

The time, \(X\) minutes, that a technician takes to fix a fault is modelled by \(X \sim N(75, 49)\). Find the probability that a repair takes more than 82 minutes.

• \(\sigma = \sqrt{49} = 7\)
• From the calculator: normal cdf with lower = 82, upper = 10^99, μ = 75, σ = 7
• P = 0.159

The volume of paint in a tin is normally distributed with mean 500 ml and standard deviation 20 ml. Sketch the normal curve and shade the region for the probability that a tin has less than 470 ml. Then find the probability.

• Sketch: bell curve centred at 500; mark 470 to the left of the centre and shade the area to its left
• From the calculator: normal cdf with lower = −10^99, upper = 470, μ = 500, σ = 20
• P = 0.0668

The box plot shows the time in minutes that 50 people spent in a shop. A student says a normal distribution is a good model. Do you agree? Give a reason.

• Median − Q1 = 4 and Q3 − median = 12
• The upper whisker (38) is much longer than the lower (4)
• The data is skewed, but a normal distribution is symmetric, so it is not a good model.

The time that Year 10 students take to run 1 500 m is normally distributed with mean 7.2 minutes and standard deviation 0.9 minutes.

1. (a) Find the probability that a student runs it in under 6 minutes.

   • From the calculator: normal cdf with lower = −10^99, upper = 6, μ = 7.2, σ = 0.9
   • P = 0.0912

2. (b) Find the probability that a student takes between 6.5 and 8 minutes.

   • From the calculator: normal cdf with lower = 6.5, upper = 8, μ = 7.2, σ = 0.9
   • P = 0.595

The thickness of a coin is modelled by \(X \sim N(2.5, 0.0016)\) mm. A coin is acceptable if its thickness is between 2.44 mm and 2.56 mm.

1. (a) Write down the standard deviation.

   • \(\sigma = \sqrt{0.0016} = 0.04\)

2. (b) Find the probability that a coin is acceptable.

   • From the calculator: normal cdf with lower = 2.44, upper = 2.56, μ = 2.5, σ = 0.04
   • P = 0.866

The diagram shows the normal distribution of the lengths of a type of leaf, in millimetres, with mean 34 and standard deviation 5. The shaded region is under the curve.

1. (a) Write the probability that the shaded region represents.

   • The shaded region lies between 30 and 40
   • \(P(30 < X < 40)\)

2. (b) Find its value.

   • From the calculator: normal cdf with lower = 30, upper = 40, μ = 34, σ = 5
   • P = 0.673

A machine fills bags of rice. The mass is normally distributed with mean 503 g and standard deviation 6 g. A bag is accepted if its mass is between 490 g and 515 g.

1. (a) Find the probability that an item is accepted.

   • From the calculator: normal cdf with lower = 490, upper = 515, μ = 503, σ = 6
   • P(accepted) = 0.962

2. (b) Find the probability that an item is rejected.

   • Rejected is the complement: 1 − P(accepted), using the unrounded value
   • P(rejected) = 0.0379
   • Check: lower tail 0.0151 + upper tail 0.0228