Statistics review Problem set B Worked solutions

State whether each is discrete or continuous data.

1. (a) The masses of bags of flour.

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

2. (b) The number of siblings a student has.

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

3. (c) The temperature at noon.

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

4. (d) The number of pages in a book.

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

The table shows the number of brothers and sisters of each student in a class.

1. (a) Write down the total frequency.

   • Add the frequencies: 6+11+7+4+2+1 = 31

2. (b) Write down the mode.

   • The highest frequency is 11, at 1. The mode is 1.

3. (c) Find the mean.

   • Enter the values and the frequencies in two lists.
   • From the calculator: mean = 1.61

4. (d) Find the standard deviation.

   • From the calculator: σ = 1.29

The box plot shows the distances, in metres, thrown by 80 athletes in a javelin competition.

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 = 55 − 35 = 20

4. (d) Find the range.

   • Range = 75 − 20 = 55

The mean delivery time, in minutes, of a pizza shop on each of 120 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 = 40 − 25 = 15

4. (d) Write down the number of days with between 25 and 30 minutes.

   • Q1 to the median is 25% of the days: 25% of 120 = 30 days

5. (e) Write down the number of days with between 25 and 55 minutes.

   • Q1 to the maximum is 75% of the days: 75% of 120 = 90 days

6. (f) On a day with a road closure the mean delivery time was 60 minutes. Determine whether this day is an outlier.

   • Lower fence: 25 − 1.5 × 15 = 2.5
   • Upper fence: 40 + 1.5 × 15 = 62.5
   • 60 is between the fences. It is not an outlier.

The table shows the number of parcels delivered to each of 48 houses on a street in one week.

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

   • The number of parcels is counted, so it is discrete.

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

   • Total: 4+8+16+12+8 = 48
   • The highest frequency is 16, at 2.

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

   • From the calculator, with the values and frequencies in lists:
   • mean = 2.25, σ = 1.16

4. (d) A stratified sample of 12 houses is taken, using the number of parcels as the strata. Find how many houses in the sample had 2 parcels.

   • Sample fraction = 12/48
   • 16 × 12/48 = 4

The number of emails received by a manager on each of 14 days was: 34, 41, 28, 52, 45, 38, 31, 49, 36, 44, 29, 95, 40, 33

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

   • From the calculator, one-variable statistics:
   • Q1 = 33, median = 39, Q3 = 45

2. (b) Find the interquartile range.

   • IQR = 45 − 33 = 12

3. (c) Show that 95 is an outlier.

   • Upper fence: 45 + 1.5 × 12 = 63
   • 95 > 63, so it is an outlier.

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

   • Remove 95: 13 values with sum 500
   • Mean = 500/13 = 38.5

A hiker records the height above sea level, x hundred metres, and the air temperature, y °C, at eight points on a mountain path. x: 2.1, 3.4, 4.5, 5.2, 6.3, 7.4, 8, 9.1 y: 11.4, 8.9, 7.2, 6.8, 3.9, 2.1, 0.8, −1.5

1. (a) State whether the air temperature is discrete or continuous.

   • Temperature is measured, so y is continuous.

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 = −0.996
   • |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 = −1.82x + 15.4

4. (d) Use the equation to estimate the air temperature when the height is 6.

   • y = −1.82 × 6 + 15.4 = 4.49

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

   • The gradient is −1.82.
   • Each extra hundred metres goes with a decrease of 1.82 °C in the air temperature.

6. (f) Explain why the line should not be used when the height is 30 hundred metres.

   • The data cover 2.1 to 9.1. 30 hundred metres is outside this, so the estimate is an extrapolation and may be unreliable.

The daily steps, in hundreds, of two walking groups, each recorded over 80 days, 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.
   • Group M 65, Group N 70

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

   • Group M: 80 − 55 = 25
   • Group N: 75 − 60 = 15

3. (c) Write down the number of days in Group M with a number of steps between 65 and 80.

   • Median to Q3 is 25% of 80 = 20 days.

4. (d) One more day is recorded for Group N with a number of steps of 100. Determine whether this is an outlier for Group N.

   • IQR for Group N = 15
   • Lower fence = 37.5, upper fence = 97.5
   • 100 is outside the fences, so it is an outlier.

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

   • IQR of Group M = 25, of Group N = 15.
   • Group M has the greater IQR, so it is more spread out in the middle half.