Statistics review Problem set A Worked solutions

State whether each is discrete or continuous data.

1. (a) The heights of seedlings.

   • Continuous: it is measured, so it can take any value in a range.

2. (b) The number of goals in a match.

   • Discrete: it is counted, so only whole-number values are possible.

3. (c) The time taken to run 100 m.

   • Continuous: it is measured, so it can take any value in a range.

4. (d) The number of rooms in a hotel.

   • Discrete: it is counted, so only whole-number values are possible.

The table shows the number of goals scored by a team in each match of a season.

1. (a) Write down the total frequency.

   • Add the frequencies: 4+9+12+7+3+1 = 36

2. (b) Write down the mode.

   • The highest frequency is 12, at 2. The mode is 2.

3. (c) Find the mean.

   • Enter the values and the frequencies in two lists.
   • From the calculator: mean = 1.97

4. (d) Find the standard deviation.

   • From the calculator: σ = 1.21

The box plot shows the times, in seconds, taken by 50 players to finish a puzzle.

1. (a) Write down the median.

   • The median is the line inside the box.

2. (b) Write down the minimum.

   • The minimum is the left end of the whisker.

3. (c) Find the interquartile range.

   • IQR = Q3 − Q1 = 70 − 45 = 25

4. (d) Find the range.

   • Range = 90 − 30 = 60

The number of visitors to a museum on each of 160 days is summarised in the box plot.

1. (a) Write down the median.

   • The median is the line in the box.

2. (b) Write down the maximum.

   • The maximum is the right end of the whisker.

3. (c) Find the interquartile range.

   • IQR = 330 − 210 = 120

4. (d) Write down the number of days with between 210 and 260 visitors.

   • Q1 to the median is 25% of the days: 25% of 160 = 40 days

5. (e) Write down the number of days with between 210 and 470 visitors.

   • Q1 to the maximum is 75% of the days: 75% of 160 = 120 days

6. (f) A storm closed most of the museum on one day and only 25 people visited. Determine whether this day is an outlier.

   • Lower fence: 210 − 1.5 × 120 = 30
   • Upper fence: 330 + 1.5 × 120 = 510
   • 25 is below the lower fence. It is an outlier.

The table shows the number of days that each of 60 students was absent last term.

1. (a) State whether the number of days absent is discrete or continuous.

   • The number of days absent is counted, so it is discrete.

2. (b) Write down the total frequency and the mode.

   • Total: 6+9+15+12+9+6+3 = 60
   • The highest frequency is 15, at 2.

3. (c) Find the mean and the standard deviation.

   • From the calculator, with the values and frequencies in lists:
   • mean = 2.65, σ = 1.62

4. (d) A stratified sample of 20 students is taken, using the number of days absent as the strata. Find how many students in the sample had 3 days absent.

   • Sample fraction = 20/60
   • 12 × 20/60 = 4

The daily rainfall, in mm, was recorded on 14 days: 2.1, 0, 3.4, 1.2, 5.6, 2.8, 0.5, 4.1, 3, 2.2, 1.5, 18.6, 3.9, 2.4

1. (a) Find the median and the lower and upper quartiles.

   • From the calculator, one-variable statistics:
   • Q1 = 1.5, median = 2.6, Q3 = 3.9

2. (b) Find the interquartile range.

   • IQR = 3.9 − 1.5 = 2.4

3. (c) Show that 18.6 is an outlier.

   • Upper fence: 3.9 + 1.5 × 2.4 = 7.5
   • 18.6 > 7.5, so it is an outlier.

4. (d) Find the mean of the values that are not outliers.

   • Remove 18.6: 13 values with sum 32.7
   • Mean = 32.7/13 = 2.52

A kiosk records the maximum temperature, x °C, and the number of bottles of water, y, sold on each of eight days. x: 18, 21, 24, 27, 30, 32, 25, 29 y: 62, 70, 88, 95, 120, 128, 85, 110

1. (a) State whether the number of bottles sold is discrete or continuous.

   • Bottles are counted, so y is discrete.

2. (b) Find Pearson's correlation coefficient r and describe the correlation.

   • Enter x and y in two lists, then linear regression:
   • From the calculator: r = 0.983
   • |r| is close to 1: strong. The sign shows the direction.

3. (c) Find the equation of the regression line of y on x.

   • From the calculator, the regression line:
   • y = 4.86x − 30.4

4. (d) Use the equation to estimate the number of bottles sold when the temperature is 26.

   • y = 4.86 × 26 − 30.4 = 96.0

5. (e) Interpret the gradient of the line.

   • The gradient is 4.86.
   • Each extra degree goes with an increase of 4.86 bottles in the number of bottles sold.

6. (f) Explain why the line should not be used when the temperature is 40 °C.

   • The data cover 18 to 32. 40 °C is outside this, so the estimate is an extrapolation and may be unreliable.

The scores of two classes in a test, 60 students in each class, are shown in the box plots.

1. (a) Write down the median of each group. Which group has the higher median?

   • Medians are the lines in the boxes.
   • Class P 60, Class Q 55

2. (b) Find the interquartile range of each group.

   • Class P: 70 − 45 = 25
   • Class Q: 70 − 40 = 30

3. (c) Write down the number of students in Class P with a score between 60 and 70.

   • Median to Q3 is 25% of 60 = 15 students.

4. (d) One more student joins Class P with a score of 10. Determine whether this is an outlier for Class P.

   • IQR for Class P = 25
   • Lower fence = 7.5, upper fence = 107.5
   • 10 is inside the fences, so it is not an outlier.

5. (e) Compare the spread of the two groups.

   • IQR of Class P = 25, of Class Q = 30.
   • Class Q has the greater IQR, so it is more spread out in the middle half.