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

Name Date

Linear regression and prediction Problem set B

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.

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 theatre records the ticket price x (€) and the number y of tickets sold for 8 different shows. x: 20, 25, 30, 35, 40, 45, 50, 55 y: 87, 76, 79, 64, 61, 52, 47, 38

    1. (a) Find the equation of the regression line of y on x.
    2. (b) Use the line to predict y when x = 42.

  2. A household records the average outside temperature x (°C) and the electricity bill y (€) for 24 months, with x from 2 to 24. The regression line of y on x is \(y=145-3.8x\).

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

  3. On 9 days, a library records the number of visitors x (in tens) and the number y of books borrowed. x: 3, 4, 6, 7, 9, 10, 12, 13, 15 y: 21, 26, 24, 30, 31, 34, 33, 2, 36

    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. The population y (thousands) of a town is recorded x years after the year 2000, for x from 5 to 20. The regression line of y on x is \(y=3.7x+41\), with r = 0.96.

    1. (a) Predict y when x = 12.
    2. (b) Predict y when x = 45. Comment on the reliability of this prediction.

In context

  1. For 10 winter days, the outside temperature x (°C) and the gas used y (m³) in a house are recorded. x: 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 y: 96, 86, 83, 73, 40, 63, 52, 50, 42, 33

    81012141618202224262830102030405060708090100x (°C)y (m³)
    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 hot drink is left to cool. Its temperature y (°C) after x minutes is: x: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 y: 80, 66, 55, 41, 29, 24, 22, 21, 18, 16

    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 = 17.

  3. A runner records the distance x (km) she trains in a week and her time y (minutes) for a 5 km race on 18 weeks. The distances ranged from 15 km to 50 km. The regression line of y on x is \(y=-0.21x+31.5\) and the mean distance was 38 km.

    1. (a) Find the mean race time.
    2. (b) She solves \(20=-0.21x+31.5\) to find the training distance needed for a time of 20 minutes. Explain why this is not a reliable use of the regression line.

Stretch

  1. A phone has x apps open, and y% of the battery left after an hour. x: 5, 6, 8, 9, 11, 12, 14, 15 y: 42, 39, 35, 33, 28, 25, 20, 18 In power-saving mode, with 16 apps open, 40% is left. Add the point (16, 40).

    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 survey of 50 teenagers finds the regression line \(y=8.9-0.35x\) for hours of sleep y on daily screen time x (hours), with x from 1 to 8 and r = −0.52.

    1. (a) Interpret the intercept. Explain why it should be treated with care.
    2. (b) Predict y when x = 5 and when x = 16. Comment on each prediction.

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