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IB Mathematics: Applications and Interpretation SL · 13.2: Probability Distributions · 13.2.8

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Normal probabilities with the calculator Problem set A

Define X, sketch if it helps, and use the calculator's normal cdf. Give probabilities to 3 significant figures.

Difficulty: first skills applying them stretch

Skills

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

    Data010203040506070Time (minutes)

In context

  1. 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.
    2. (b) Find the probability that a student takes between 6.5 and 8 minutes.

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

Stretch

  1. 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.

    10152025303540455055Length (mm)
    1. (a) Write the probability that the shaded region represents.
    2. (b) Find its value.

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

jmaths.xyz · AI SL · 13.2.8 Normal probability calculations with the GDC