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Math lesson plan · 12th grade (Year 13 in England) · 30 minutes

Find normal probabilities with z-scores and technology

Statistics and probability

CCSS HSS.ID.A.4England: A-level Statistics

Objective and success criteria

Students will: Find normal probabilities with z-scores and technology.

  • I can answer questions on today's skill on my own, without help.
  • I can explain each step and say why it works.
  • I can check my answer and spot a mistake.

Before and after

Needs first: Combine regions of a normal curve with the 68–95–99.7 rule; Calculate and use z-scores

Leads to: the next skill in this area.

Materials: Squared paper for charts, Class survey data, Sticky notes for a live chart, Mini whiteboards and pens, Printed student sheet (below).

Warm-up · 4 min

  • Quick retrieval from earlier learning (Combine regions of a normal curve with the 68–95–99.7 rule; Calculate and use z-scores). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. A test has mean 79 and standard deviation 11. Find the z-score of a mark of 62.5. → -1.5Prompt: z = (x − mean) ÷ standard deviation.
  2. A quantity is normally distributed with mean 76 and standard deviation 8. What percentage of values lie between 84 and 92? → 13.5Prompt: Convert each end to 'number of standard deviations from the mean'.
  3. A factory fills 800 bottles. Volumes are normally distributed with mean 110 ml and standard deviation 24 ml. About how many bottles hold less than 62 ml? → 20Prompt: That value is 2 standard deviations from the mean.

Hook · 3 min

  • Take a quick class survey (favorite fruit, travel to school) and ask what the results tell us.
  • Share the objective: Find normal probabilities with z-scores and technology. Read the success criteria together.

I do: model it · 7 min

  • Work through the example out loud, saying why you do each step, not just what you do.
  • Example: X is normally distributed with mean 153 and standard deviation 20. Find P(X < 134.8) to 3 decimal places.
  • Step 1: z = (134.8 − 153)/20 ≈ −0.91
  • Step 2: P ≈ 0.181
  • Answer: 0.181
  • Pause after each step and ask a student to say what comes next.

We do: guided practice · 6 min

  • Do these together. Students try each step on whiteboards before you reveal it.
  • Use the hints as prompts rather than giving the answer.
  1. X is normally distributed with mean 157 and standard deviation 5. Find P(X < 165.1) to 3 decimal places. → 0.947Prompt: Standardise: z = (x − μ)/σ.
  2. Battery life is normally distributed with mean 375 minutes and standard deviation 33 minutes. Find the probability that a battery lasts between 362 and 408 minutes, to 3 decimal places. → 0.495Prompt: P(a < X < b) = P(X < b) − P(X < a).

You do: independent practice · 7 min

  • Students work on the student sheet. Circulate and check the first two answers of every student early.
  • Students who finish: write a question of their own for a partner, with the answer.

The 6 questions are on the student sheet below, with answers in the key.

Exit ticket · 3 min

  • Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
  1. Battery life is normally distributed with mean 422 minutes and standard deviation 35 minutes. Find the probability that a battery lasts between 369 and 447 minutes, to 3 decimal places. → 0.697Prompt: P(a < X < b) = P(X < b) − P(X < a).
  2. X is normally distributed with mean 87 and standard deviation 17. Find P(X < 59.8) to 3 decimal places. → 0.055Prompt: Standardise: z = (x − μ)/σ.

Watch out for

  • Forgetting to put data in order before finding the median.
  • Reading a bar's height wrongly when the scale goes up in 2s or 5s.

Support and stretch

Support

  • Review first: Combine regions of a normal curve with the 68–95–99.7 rule.
  • Use concrete materials: squared paper for charts.
  • Give the first step of each question, or a partly worked example to finish.
  • Pair with a confident partner for the guided practice.

Stretch

  • Explain a common mistake in writing and how to fix it.
  • Write a word problem that uses today's skill, then solve a partner's.

Homework and more practice

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-12.

Free lesson plan from eurekami.com/lesson-plan-builder

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Find normal probabilities with z-scores and technology

Name

Date

Score/ 6

  1. Question 1

    Battery life is normally distributed with mean 177 minutes and standard deviation 21 minutes. Find the probability that a battery lasts between 172 and 211 minutes, to 3 decimal places.

    Answer

  2. Question 2

    Adult heights are normally distributed with mean 184 cm and standard deviation 5 cm. Find P(height > 186 cm) to 3 decimal places.

    Answer

  3. Question 3

    X is normally distributed with mean 55 and standard deviation 9. Find P(X < 60.6) to 3 decimal places.

    Answer

  4. Question 4

    Battery life is normally distributed with mean 167 minutes and standard deviation 40 minutes. Find the probability that a battery lasts between 143 and 226 minutes, to 3 decimal places.

    Answer

  5. Question 5

    Adult heights are normally distributed with mean 153 cm and standard deviation 11 cm. Find P(height > 174 cm) to 3 decimal places.

    Answer

  6. Question 6

    X is normally distributed with mean 102 and standard deviation 17. Find P(X < 86.4) to 3 decimal places.

    Answer

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Answer key: Find normal probabilities with z-scores and technology

  1. 1. 0.541

  2. 2. 0.345

  3. 3. 0.733

  4. 4. 0.656

  5. 5. 0.028

  6. 6. 0.179