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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 B

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

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. The time a swimmer takes to swim 100 m is normally distributed with mean 64.0 s and standard deviation 2.8 s. Find the probability that she takes more than 68 s.

  2. The time to download a file is normally distributed with mean 7.5 seconds and standard deviation 2.1 seconds. Find the probability that a download takes less than 5 seconds.

  3. The number of steps an office worker walks in a day is normally distributed with mean 8 200 and standard deviation 1 400. Find the probability that a worker walks between 7 000 and 10 000 steps in a day.

  4. The mass, \(X\) grams, of a pebble from a beach is modelled by \(X \sim N(40, 25)\). Find the probability that a pebble has mass less than 33 g.

  5. The length of a bolt is normally distributed with mean 18 mm and standard deviation 2.5 mm. Sketch the normal curve and shade the region for the probability that a bolt is shorter than 14 mm. Then find the probability.

  6. The box plot shows the number of goals scored in a season by 70 amateur players. A student says a normal distribution is a good model. Do you agree? Give a reason.

    Data0510152025Goals scored

In context

  1. The time that members of a cycling club take to ride 20 km is normally distributed with mean 41 minutes and standard deviation 5 minutes.

    1. (a) Find the probability that a rider takes less than 35 minutes.
    2. (b) Find the probability that a rider takes between 38 and 50 minutes.

  2. The internal diameter of a pipe is modelled by \(X \sim N(20, 0.09)\) mm. A pipe is acceptable if its diameter is between 19.5 mm and 20.5 mm.

    1. (a) Write down the standard deviation.
    2. (b) Find the probability that a pipe is acceptable.

Stretch

  1. The diagram shows the normal distribution of the daily number of visitors to a museum, with mean 110 and standard deviation 12. The shaded region is under the curve.

    60728496108120132144156168Visitors
    1. (a) Write the probability that the shaded region represents.
    2. (b) Find its value.

  2. A machine fills bottles of milk. The volume is normally distributed with mean 1 015 ml and standard deviation 9 ml. A bottle is accepted if its volume is between 1 000 ml and 1 030 ml.

    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