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IB Mathematics: Applications and Interpretation SL · 13.3: Statistics · 13.3.7

Name Date

Linear regression and prediction Problem set A

Use the calculator for the regression line, r and quartiles. Give values to 3 significant figures. Say whether a prediction is inside the data range.

Difficulty: first skills applying them stretch

Skills

  1. A kiosk records the hours of sunshine x in a day and the number y of cold drinks sold on 8 days. x: 3, 5, 6, 8, 9, 11, 12, 14 y: 14, 19, 20, 27, 27, 33, 36, 41

    1. (a) Find the equation of the regression line of y on x.
    2. (b) Use the line to predict y when x = 10.

  2. A taxi firm models the fare y (€) for a journey of x km. The regression line of y on x, from 30 journeys of 2 km to 25 km, is \(y=1.9x+4.5\).

    1. (a) Interpret the gradient.
    2. (b) Interpret the intercept.

  3. On 10 afternoons, a shop records the number of customers x and the number of items sold y. x: 12, 15, 16, 18, 19, 21, 22, 24, 25, 27 y: 40, 46, 45, 52, 49, 58, 55, 61, 95, 66

    1. (a) Find the quartiles of y. Use the 1.5 × IQR rule to test y for outliers.
    2. (b) Test x for outliers.

  4. A sunflower's height y (cm) is measured x weeks after planting, for x from 5 to 25. The regression line of y on x is \(y=6.4x+12\), with r = 0.97.

    1. (a) Predict y when x = 15.
    2. (b) Predict y when x = 60. Comment on the reliability of this prediction.

In context

  1. For 10 students, the weekly revision time x (hours) and the exam mark y (%) are recorded. x: 2, 4, 5, 7, 8, 10, 11, 13, 14, 16 y: 38, 44, 50, 52, 35, 66, 68, 77, 80, 88

    24681012141618102030405060708090100x (hours)y (%)
    1. (a) Write down the coordinates of the point that does not fit the pattern.
    2. (b) Find the regression line of y on x with and without this point.
    3. (c) Describe the effect of the point on the line and on r.

  2. A trainee's typing speed y (words per minute) after x weeks on a course is: x: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 y: 3, 6, 10, 13, 16, 17, 18, 18, 20, 22

    1. (a) Plot the data. Explain why one line is not a good model.
    2. (b) Find the regression line for each part.
    3. (c) Predict y when x = 8.5.

  3. A café records the midday temperature x (°C) and the daily sales y (€) on 30 days. The temperatures ranged from 9 °C to 30 °C. The regression line of y on x is \(y=18.5x+120\) and the mean temperature was 21.4 °C.

    1. (a) Find the mean daily sales.
    2. (b) The owner solves \(700=18.5x+120\) to find the temperature needed for sales of €700. Explain why this is not a reliable use of the regression line.

Stretch

  1. In week x of a fitness programme, a client does y press-ups. x: 1, 2, 3, 4, 5, 6, 7, 8 y: 12, 15, 17, 22, 24, 27, 31, 33 In week 9 the client does 15 press-ups, after a minor injury. Add the point (9, 15).

    1. (a) Find the regression line and r for the original data.
    2. (b) Find the regression line and r when the new point is included.
    3. (c) Describe the effect of the new point.

  2. A study of 40 adults finds the regression line \(y=78-1.7x\) for resting heart rate y (beats per minute) on weekly exercise x (hours), with x from 1 to 9 and r = −0.46.

    1. (a) Interpret the intercept. Explain why it should be treated with care.
    2. (b) Predict y when x = 6 and when x = 20. Comment on each prediction.

jmaths.xyz · AI SL · 13.3.7 Linear regression and prediction