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IEB practice · Paper II

IEB Statistics practice

6 newly generated questions across all 13 statistics topics the IEB examines in Paper II, 69 marks in all. Work each one, then open its answer underneath to check yourself. Nothing here is taken from a past paper.

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The least-squares line of best fit

1 questions · 10 marks

110 marks

The table and scatter plot show the number of absences in a term (x) and a learner's mark out of 100 (y): (0; 100); (4; 81); (6; 76); (9; 62); (10; 53); (11; 52); (13; 48); (14; 45).

−20246810121416405060708090100110
  1. (a)
    Use your calculator to determine the equation of the least squares regression line in the form y = A + Bx. Give A and B correct to three decimal places.
    (3)
  2. (b)
    Determine the correlation coefficient r, correct to two decimal places.
    (1)
  3. (c)
    Describe the strength and direction of the correlation.
    (2)
  4. (d)
    Use your equation to predict y when x = 5.
    (2)
  5. (e)
    Is this prediction reliable? Explain.
    (2)
Show the answers

(a) Answer: y = 98.357 − 4.028x

  1. A = 98.357
  2. B = −4.028

(b) Answer: r = −0.99

  1. r = −0.99

(c) Answer: a strong negative correlation

  1. r = −0.99: r < 0 and |r| is at least 0.8

(d) Answer: y = 78.22

  1. y = 98.357 − 4.028(5)
  2. y = 78.22

(e) Answer: Yes, reasonably: x lies within the data (interpolation)

  1. the data runs from x = 0 to x = 14; x = 5 lies inside it

The estimated mean of grouped data

2 questions · 25 marks

215 marks

The frequency polygon shows the mass in kg of 49 parcels, grouped in the classes 0 ≤ x < 5; 5 ≤ x < 10; 10 ≤ x < 15; 15 ≤ x < 20; 20 ≤ x < 25.

−505101520253005101520
  1. (a)
    Estimate the mean of the data.
    (3)
  2. (b)
    Write down the modal class.
    (1)
  3. (c)
    Draw an ogive (cumulative frequency curve) for the data.
    (4)
  4. (d)
    Use your ogive to estimate the median.
    (2)
  5. (e)
    Use your ogive to estimate the interquartile range.
    (3)
  6. (f)
    Is the data skewed? Explain, using the mean and the median.
    (2)
Show the answers

(a) Answer: 14.74

  1. midpoints: 2.5, 7.5, 12.5, 17.5, 22.5
  2. Σ f·x = 722.5 and n = 49
  3. mean = 722.5/49 = 14.74

(b) Answer: 20 ≤ x < 25

  1. the highest frequency is 18

(c) Answer: See the ogive below.

051015202501020304050
  1. cumulative frequencies: 8, 10, 27, 31, 49
  2. plot each against the upper limit of its class, starting at (0; 0)

(d) Answer: about 14.3

  1. the median is at a cumulative frequency of 49/2 = 24.5
  2. read across to the ogive and down: about 14.3

(e) Answer: about 10.9

  1. Q1 at 12.25: about 10.7; Q3 at 36.75: about 21.6
  2. IQR = 21.6 − 10.7 = 10.9

(f) Answer: Skewed to the right (positively)

  1. mean 14.74 and median about 14.3
  2. mean > median
310 marks

The table shows the time in minutes that 40 people waited at a clinic. One frequency is unknown.

  • 0 ≤ x < 10: frequency 15
  • 10 ≤ x < 20: frequency 11
  • 20 ≤ x < 30: frequency 4
  • 30 ≤ x < 40: frequency 5
  • 40 ≤ x < 50: frequency x
  1. (a)
    The estimated mean is 18.5. Determine the value of x.
    (3)
  2. (b)
    Write down the modal class.
    (1)
  3. (c)
    In which class does the median lie?
    (2)
  4. (d)
    Complete the cumulative frequencies and draw the ogive.
    (4)
Show the answers

(a) Answer: x = 5

  1. (515 + 45x)/(35 + x) = 18.5
  2. 515 + 45x = 18.5(35 + x)
  3. x = 5

(b) Answer: 0 ≤ x < 10

  1. the highest frequency is 15

(c) Answer: 10 ≤ x < 20

  1. n = 40: the median lies between the 20th and 21st values
  2. cumulative frequencies: 15, 26, 30, 35, 40
  3. 10 ≤ x < 20

(d) Answer: See the ogive below.

010203040500510152025303540
  1. cumulative frequencies: 15, 26, 30, 35, 40
  2. plot each against the upper limit of its class, starting at (0; 0)

Scatter plots

1 questions · 9 marks

49 marks

The scatter plot shows the midday temperature in °C (x) and the number of ice creams a shop sold (y): (15; 30); (16; 32); (21; 63); (22; 68); (27; 92); (29; 109); (30; 71); (33; 126); (34; 138).

10152025303520406080100120140160
  1. (a)
    Identify the outlier in the data.
    (1)
  2. (b)
    The outlier is removed. Determine the equation of the least squares regression line for the remaining data, in the form y = A + Bx, with A and B correct to three decimal places.
    (3)
  3. (c)
    Determine the correlation coefficient r of the remaining data, correct to two decimal places.
    (1)
  4. (d)
    By how much does y change, on average, for each increase of 1 in x? Use your equation.
    (2)
  5. (e)
    Did the outlier make the correlation stronger or weaker? Explain.
    (2)
Show the answers

(a) Answer: (30; 71)

  1. (30; 71) lies far from the trend of the other points

(b) Answer: y = −55.747 + 5.604x

  1. A = −55.747
  2. B = 5.604

(c) Answer: r = 1.00

  1. r = 1.00

(d) Answer: y increases by about 5.604

  1. B = 5.604 is the change in y for each 1 added to x

(e) Answer: Weaker

  1. with the outlier r = 0.94; without it r = 1.00
  2. |r| is smaller with the outlier: it weakened the correlation

Mean, median and mode

2 questions · 25 marks

515 marks

The data below gives the daily rainfall in mm on 12 days: 8; 14; 15; 15; 16; 17; 20; 21; 21; 25; 28; 31

  1. (a)
    Calculate the mean.
    (1)
  2. (b)
    Calculate the standard deviation, correct to two decimal places.
    (2)
  3. (c)
    How many of the values lie within one standard deviation of the mean?
    (2)
  4. (d)
    Write down the five-number summary.
    (3)
  5. (e)
    Draw a box-and-whisker plot of the data.
    (2)
  6. (f)
    An outlier is a value more than 1.5 × IQR above Q3 or below Q1. Is 31 an outlier? Show your working.
    (3)
  7. (g)
    Every value is increased by 10. What happens to the mean and to the standard deviation?
    (2)
Show the answers

(a) Answer: 19.25

  1. Σx = 231 and n = 12
  2. mean = 231/12 = 19.25

(b) Answer: σ = 6.19

  1. σ² = Σ(x − 19.25)²/12 = 38.35
  2. σ = 6.19

(c) Answer: 9

  1. between 13.06 and 25.44
  2. 9 values

(d) Answer: 8; 15; 18.5; 23; 31

  1. minimum 8; Q1 15; median 18.5; Q3 23; maximum 31

(e) Answer: See the plot below.

510152025303581518.52331
  1. the box from Q1 to Q3 with the median marked; whiskers to the minimum and maximum

(f) Answer: No: 31 ≤ 35

  1. IQR = 23 − 15 = 8
  2. Q3 + 1.5 × IQR = 23 + 12 = 35
  3. 31 ≤ 35

(g) Answer: The mean increases by 10; the standard deviation stays the same.

  1. every value, and so their mean, moves by the same amount
  2. the distances from the mean do not change
610 marks

The 7 values below give the daily rainfall in mm on 7 days, in ascending order: 5; x + 4; x + 5; x + 10; x + 18; x + 19; x + 22.

  1. (a)
    The median is 29. Determine the value of x.
    (2)
  2. (b)
    Calculate the mean.
    (2)
  3. (c)
    Calculate the standard deviation, correct to two decimal places.
    (2)
  4. (d)
    How many of the values lie more than one standard deviation above the mean?
    (2)
  5. (e)
    The highest value is increased by 10. Which of the mean, the median and the standard deviation change?
    (2)
Show the answers

(a) Answer: x = 19

  1. n = 7: the median is the 4th value, x + 10
  2. x + 10 = 29, so x = 19

(b) Answer: 28.14

  1. the data: 5; 23; 24; 29; 37; 38; 41
  2. mean = 197/7 = 28.14

(c) Answer: 11.47

  1. σ = 11.47 (calculator)

(d) Answer: 1

  1. mean + σ = 28.14 + 11.47 = 39.61
  2. 1 value(s) above it

(e) Answer: The mean and the standard deviation; the median stays the same

  1. the total grows by 10, so the mean grows
  2. the top value moves further from the rest, so σ grows
  3. the middle value does not move

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The same questions as a booklet to print, with the memorandum as a separate file so you can work without the answers in front of you. In English and Afrikaans.

Every question on this page was generated for this site by a computer engine, and its answer was confirmed by an independent method at the moment the question was made.

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