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IB Mathematics: Applications and Interpretation SL · 13.2: Probability Distributions · 13.2.7

Name Date

The normal distribution and when to use it Problem set B

Use the 68–95–99.7 rule and your reading of the diagram. Give reasons in a sentence.

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. A pinball toy has 14 rows of pegs. At each peg a marble goes left or right with probability 0.5, independently. Let B be the number of times a ball goes right, so \(B \sim B(14, 0.5)\).

    1. (a) Find the mean and standard deviation of B.
    2. (b) Many balls are dropped. Describe the shape of the pile of balls and explain why it has that shape.

  2. The volume of juice in a bottle is normally distributed with mean 250 ml and standard deviation 12 ml. Use the 68–95–99.7 rule.

    1. (a) Find the percentage between 226 and 274 ml.
    2. (b) Find the percentage below 238 ml.
    3. (c) Find the percentage between 250 and 274 ml.

  3. The box plot shows the number of messages that 90 students sent in one day. Is a normal distribution a suitable model? Give reasons.

    Data051015202530Messages sent

  4. The box plot shows the times in seconds of 120 runners in a 100 m race. Is a normal distribution a suitable model? Give a reason.

    Data101214161820222426Time (s)

In context

  1. The number of centimetres of rain that fall in a day in a desert town is modelled by a normal distribution with mean 1.6 and standard deviation 1.1.

    1. (a) How many standard deviations is the limit 0 below the mean?
    2. (b) Explain why a normal distribution is not a good model. Use your answer to part (a).

Stretch

  1. The diameter of a piston is normally distributed with mean 80 mm and standard deviation 0.4 mm.

    1. (a) Use the 68–95–99.7 rule to give an interval that contains about 99.7% of the values.
    2. (b) One piston measures 81.5 mm. How many standard deviations from the mean is that? Is it unusual?

  2. A pinball toy is tilted so that a marble goes right with probability 0.15 at each of its 12 rows of pegs. Let B be the number of times a ball goes right, so \(B \sim B(12, 0.15)\).

    1. (a) Find the mean and standard deviation of B.
    2. (b) How many standard deviations is 0 below the mean?
    3. (c) Explain why the pile of balls is not a normal shape.

jmaths.xyz · AI SL · 13.2.7 The normal distribution: properties and when it is a good model