A Level Math Revision Free diagnostic
Themed maths

Big tournament statistics for A Level

A ready-to-teach statistics lesson for the summer term, built on a made-up tournament: a 5-minute starter, a 35-minute main activity on shots and goals, penalty shoot-outs and race times, an extension and full worked answers. All the data are made up for this lesson.

Level
A Level Maths (Year 12 and Year 13)
Time
40 minutes, plus a 10-minute extension
Topics
Correlation and regression; Hypothesis test for correlation (Year 13); The binomial distribution; The normal distribution (Year 13)
Equipment
The starter is non-calculator. A calculator with statistics functions is needed.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A12 minShots and goals
Main: task B11 minPenalty shoot-out
Main: task C12 minRace times
Extension10 minFast finishers or homework

Starter (5 minutes)

No calculator.

  1. Find the mean of 2, 4, 4, 5, 10.
  2. Ten values have Σx = 146. Find the mean.
  3. X ~ B(4, ½). Find P(X = 2).
  4. Z ~ N(0, 1). Write down P(Z < 0).

Main activity (35 minutes)

Task A: Shots and goals (12 min)

Made-up data for ten teams. Shots on target x: 12, 15, 9, 20, 17, 11, 14, 22, 8, 18. Goals y: 4, 5, 2, 8, 6, 3, 5, 9, 2, 6.

  1. Find the product moment correlation coefficient, to 3 significant figures.
  2. Find the regression line of y on x, with coefficients to 3 significant figures, and interpret the gradient.
  3. (Year 13) Test at the 5% level whether there is positive correlation between shots on target and goals. The critical value for n = 10 (one tail, 5%) is 0.5494.

Task B: Penalty shoot-out (11 min)

In a made-up shoot-out, each of team A’s five takers scores with probability 0.8 and each of team B’s with probability 0.7, independently.

  1. Find the probability that team A scores all 5 penalties, to 3 significant figures.
  2. Find the probability that team A scores at least 4, to 3 significant figures.
  3. Each team takes all five penalties. Find the probability that team A scores more than team B, to 3 significant figures.

Task C: Race times (12 min)

(Year 13) Made-up 100 m times are modelled by T ~ N(10.2, 0.152) seconds.

  1. Find P(T < 10.0), to 3 significant figures.
  2. Find the time that only the fastest 5% beat, to 3 significant figures.
  3. Eight sprinters run independently. Find the probability that at least one runs under 10.0 s.

Extension (10 minutes)

Year 13 stretch.

  1. A coach thinks the mean is 10.2 s but only 10% of times are under 10.0 s. Find the standard deviation this implies, to 3 significant figures.

For teachers

Teacher notes and full worked answers
  • Every number in this pack is made up for the lesson; say so when you show it.
  • In task A (c), check that students write hypotheses about ρ, the population correlation, not r.
  • Task B (c) is a good spreadsheet task: a 6 by 6 table of joint probabilities.

Starter

  1. 5
    • 25 ÷ 5
  2. 14.6
    • 146 ÷ 10
  3. 3/8
    • 4C2 × (½)4 = 6/16 = 3/8
  4. 0.5
    • The distribution is symmetrical about 0.

Task A: Shots and goals

  1. 0.989
    • Calculator: r = 0.98894…
  2. y = 0.499x − 2.29; about one extra goal for every two extra shots on target
    • Calculator: y = 0.49898…x − 2.2851…
    • The gradient 0.499 means each extra shot on target goes with about 0.5 extra goals.
  3. Reject H0: there is evidence of positive correlation
    • H0: ρ = 0; H1: ρ > 0.
    • 0.989 > 0.5494, so reject H0.

Task B: Penalty shoot-out

  1. 0.328
    • 0.85 = 0.32768
  2. 0.737
    • P(4) + P(5) = 5 × 0.84 × 0.2 + 0.85 = 0.4096 + 0.32768 = 0.73728
  3. 0.501
    • Add P(A = i) × P(B = j) over all i > j, with A ~ B(5, 0.8) and B ~ B(5, 0.7).
    • The total is 0.50083…

Task C: Race times

  1. 0.0912
    • Standardise: z = (10.0 − 10.2)/0.15 = −1.333…; P(Z < −1.333…) = 0.0912
  2. 9.95 s
    • z = −1.6449; 10.2 − 1.6449 × 0.15 = 9.953…
  3. 0.535
    • 1 − (1 − 0.0912…)8 = 0.5347…

Extension

  1. 0.156 s
    • P(T < 10.0) = 0.1 gives (10.0 − 10.2)/σ = −1.2816.
    • σ = 0.2 ÷ 1.2816 = 0.1561…

The same answers are in the answers PDF. Every answer was recomputed by computer before publishing.

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