Statistics: exam review Exam review Worked solutions

A school has 600 students: 240 in Year 10, 180 in Year 11, 120 in Year 12 and 60 in Year 13. A survey needs a sample of 40 students. The sample has students from each year group in proportion to the size of the year group, chosen at random within each year group. [6]

1. (a) Name the type of sampling used. [1]

   • A1 for stratified (the sample is split into groups and each is sampled in proportion)

2. (b) Find the number of Year 12 students in the sample. [2]

   • M1 for \(\dfrac{120}{600}\times 40\)
   • A1 for 8

3. (c) The survey records (i) the number of messages a student sends in one day, and (ii) the time a student spends travelling to school. State whether each variable is discrete or continuous. [2]

   • A1 for (i) discrete (it is counted)
   • A1 for (ii) continuous (it is measured)

4. (d) A student's travel time is recorded to the nearest minute as 35 minutes. Write down the interval of values that the actual time can have. [1]

   • A1 for \(34.5\le t<35.5\)

The table shows the number of goals scored in each of 38 football matches. The frequency for 2 goals is \(k\). [7]

1. (a) Find the value of \(k\). [2]

   • M1 for \(29+k=38\)
   • A1 for \(k=9\)

2. (b) Write down the mode. [1]

   • A1 for 1 (the value with the greatest frequency)

3. (c) Use your calculator to find the mean. [2]

   • M1 for entering the values and frequencies and using one-variable statistics
   • A1 for mean \(=1.95\)

4. (d) Use your calculator to find the population standard deviation. [1]

   • A1 for \(\sigma=1.30\)

5. (e) Find the number of matches in which more than 2 goals were scored. [1]

   • A1 for 13

The four numbers \(6\), \(x\), \(10\) and \(2x\) have a mean of 10. [6]

1. (a) Find the value of \(x\). [2]

   • M1 for \(6+x+10+2x=4\times10\)
   • A1 for \(x=8\)

2. (b) The daily maximum temperatures in a city in July, in °C, have mean 18.4 and standard deviation 2.5. Each temperature \(C\) is converted to °F using \(F=1.8C+32\). Find the new mean and the new standard deviation. [2]

   • A1 for new mean \(1.8\times18.4+32=65.1\)
   • A1 for new standard deviation \(1.8\times2.5=4.5\) (the constant does not change the spread)

3. (c) The original median is 18 and the original interquartile range is 4. Find the new median and the new interquartile range. [2]

   • A1 for new median \(1.8\times18+32=64.4\)
   • A1 for new IQR \(1.8\times4=7.2\)

The waiting times, in minutes, of 6 patients at a clinic are 18, 11, 20, 15, 12, 8. [7]

1. (a) Find the mean. [1]

   • A1 for mean \(=14\)

2. (b) Show how to find the population standard deviation without using the statistics menu of your calculator. [3]

   • M1 for the deviations \(4, -3, 6, 1, -2, -6\)
   • M1 for the squares summed: \(102\), then \(\dfrac{102}{6}=17\)
   • A1 for \(\sigma=\sqrt{17}=4.12\)

3. (c) A second clinic has waiting times with a standard deviation of 2.6 minutes. State which clinic has the more consistent waiting times, and give a reason. [1]

   • A1 for the second clinic, because its standard deviation is smaller (2.6 compared with 4.12)

4. (d) The times for a third clinic are approximately normally distributed with mean 25 minutes and standard deviation 4 minutes. Use the 68–95–99.7 rule to find the interval that contains approximately 95% of the times. [2]

   • M1 for using the interval \(\mu\pm2\sigma\), about 95% of the data
   • A1 for 17 to 33

A food stall records the number of customers on each of 11 days: 42, 41, 87, 60, 58, 44, 47, 52, 56, 45, 40. [8]

1. (a) Find the median, the lower quartile and the upper quartile. [3]

   • A1 for median \(=47\)
   • A1 for \(Q_1=42\) (median of the lower 5 values)
   • A1 for \(Q_3=58\) (median of the upper 5 values)

2. (b) Find the interquartile range. [1]

   • A1 for \(58-42=16\)

3. (c) Show that the greatest value, 87, is an outlier. [2]

   • M1 for \(Q_3+1.5\times\text{IQR}=58+1.5\times16=82\)
   • R1 for 87 \(>82\), so it is an outlier

4. (d) The outlier is removed. Find the mean and the median of the remaining 10 values, and use them to explain whether the mean or the median is the better measure of the centre of the original data. [2]

   • A1 for the mean \(48.5\) (it was \(52\)) and the median \(46\) (it was \(47\))
   • R1 for the median changes less than the mean, so the median is not pulled by the outlier and is the better measure

Two couriers, A and B, are compared. The box plots show the delivery times, in minutes, of a sample of deliveries by each courier. [7]

1. (a) Write down the median for Courier A. [1]

   • A1 for 25

2. (b) Find the interquartile range for each. [2]

   • A1 for Courier A: \(31-20=11\)
   • A1 for Courier B: \(29-22=7\)

3. (c) Compare the delivery times of the two couriers. Make one comment about the centre and one about the spread. [2]

   • A1 for a comparison of the IQRs (11 and 7): Courier B has the smaller spread
   • A1 for a comparison of the medians: Courier A 25, Courier B 26

4. (d) Courier B makes a new delivery that takes 47 minutes. Determine whether this delivery time is an outlier for Courier B. Show your working. [2]

   • M1 for the fences \(Q_1-1.5\times\text{IQR}=11.5\) and \(Q_3+1.5\times\text{IQR}=39.5\)
   • R1 for 47 outside the fences, so it is an outlier

The histogram shows the masses of a sample of tomatoes. [5]

1. (a) Find the total number of tomatoes in the sample. [1]

   • A1 for 6+14+22+12+6=60

2. (b) Estimate the number of tomatoes with mass between 27 g and 38 g. [2]

   • M1 for taking part of the classes at each end in proportion to the width
   • A1 for 37.6

3. (c) Estimate the mean mass. [2]

   • M1 for using the midpoints 22.5, 27.5, 32.5, 37.5, 42.5
   • A1 for mean \(\approx32.3\)

In class X, \(25\) students sit a test and their mean mark is \(72\). In class Y, \(15\) students sit the same test and their mean mark is \(64\). [4]

1. (a) Find the mean mark of all 40 students. [2]

   • M1 for \(25\times72+15\times64=2760\), the total of all the marks, divided by \(40\)
   • A1 for \(2760\div40=69\)

2. (b) A mark in class X was entered as \(58\) but should have been \(85\). Find the correct mean mark of all 40 students. [2]

   • M1 for the total changing by \(85-58=27\), to \(2787\)
   • A1 for \(2787\div40=69.7\)

A library records the number of books borrowed on each of its visits by members. The table shows the frequencies for one afternoon. The mean number of books borrowed is \(2.8\). [5]

1. (a) Find the value of \(y\). [3]

   • M1 for the total frequency \(22+y\) and the total \(61+3y\)
   • M1 for \(\dfrac{61+3y}{22+y}=2.8\)
   • A1 for \(y=3\)

2. (b) Find the number of visits on which at most 2 books were borrowed. [1]

   • A1 for \(4+9=13\)

3. (c) Write down the mode. [1]

   • A1 for 2 (the value with the greatest frequency)

The waiting times of 80 patients at a hospital are recorded. The cumulative frequency graph shows the results. [14]

1. (a) Use the graph to estimate the median. [2]

   • M1 for reading across from a cumulative frequency of \(40\)
   • A1 for 30 (accept 29 to 31)

2. (b) Use the graph to estimate the lower quartile and the upper quartile, and find the interquartile range. [3]

   • M1 for reading at \(20\) and \(60\)
   • A1 for \(Q_1\approx20.9\)
   • A1 for \(Q_3\approx38.3\), so IQR \(\approx17.4\) (accept answers within 1 of these)

3. (c) Estimate the number of patients with waiting time greater than 45 minutes. [2]

   • M1 for reading the cumulative frequency at 45: about 70
   • A1 for \(80-70=10\) (accept 7.6 to 12.4)

4. (d) The longest 10% of waiting times are reviewed by the manager. Estimate the shortest waiting time that is reviewed. [2]

   • M1 for reading across from a cumulative frequency of \(72\)
   • A1 for 46.7 (accept 45.7 to 47.7)

5. (e) Complete a grouped frequency table for the data. [2]

   • M1 for the differences of the cumulative frequencies
   • A1 for 6, 12, 22, 24, 12, 4

6. (f) Hence estimate the mean waiting time. [3]

   • M1 for the midpoints 5, 15, 25, 35, 45, 55
   • M1 for \(\dfrac{\sum fx}{80}\)
   • A1 for \(29.5\)

Nine runners record the number of hours they train each week, \(x\), and their time for a 5 km race, \(y\) minutes. The hours of training are 1, 2, 3, 4, 5, 6, 7, 8, 9 and the times are 30.5, 32.7, 31.5, 28.3, 28.5, 27.4, 27.4, 27.1, 23.1. [14]

1. (a) Write down the coordinates of the mean point. [1]

   • A1 for \((5, 28.5)\)

2. (b) Find Pearson's product-moment correlation coefficient, \(r\). [2]

   • M1 for entering the two lists and finding the correlation coefficient
   • A1 for \(r=-0.894\)

3. (c) Describe the correlation between the number of hours of training and the 5 km time. [1]

   • A1 for strong and negative (\(r=-0.894\))

4. (d) Find the equation of the regression line of \(y\) on \(x\). [2]

   • M1 for using linear regression
   • A1 for \(y=-0.925x+33.1\)

5. (e) Interpret the gradient and the \(y\)-intercept of the regression line in the context of the question. [3]

   • A1 for the gradient: each extra hour of training per week reduces the predicted time by 0.925 minutes
   • A1 for the intercept: the predicted time with no training is 33.1 minutes
   • A1 for the gradient and the intercept each stated in context, with units

6. (f) Use the regression line to estimate the 5 km time of a runner who trains for 5.5 hours per week. [2]

   • M1 for substituting \(x=5.5\)
   • A1 for 28.0

7. (g) Ayesha uses the regression line to estimate the 5 km time of a runner who trains for 15 hours per week. Write down the estimate and comment on its reliability. [2]

   • A1 for 19.2
   • R1 for unreliable: 15 is outside the range of the data (1 to 9), so this is extrapolation

8. (h) A coach says that the correlation shows that training for more hours causes a faster 5 km time. Comment on this claim. [1]

   • R1 for correlation does not show that one variable causes the other, with a possible other factor named (such as fitness or age)

A researcher records the age, in months, of eight toddlers: 10, 12, 14, 16, 18, 20, 22, 24, and the number of words each toddler can say: 3, 5, 12, 12, 28, 45, 80, 150. The scatter diagram shows the data. [14]

1. (a) Write down the ranks of the ages and of the numbers of words, giving tied values their average rank. [3]

   • M1 for ranking each variable from smallest to largest
   • A1 for the ranks of the ages: 1, 2, 3, 4, 5, 6, 7, 8
   • A1 for the ranks of the numbers of words: 1, 2, 3.5, 3.5, 5, 6, 7, 8 (the tied values share the average rank)

2. (b) Find Spearman's rank correlation coefficient, \(r_s\). [2]

   • M1 for using the correlation of the ranks (or the calculator's rank correlation)
   • A1 for \(r_s=0.994\)

3. (c) Interpret the value of \(r_s\) in the context of the question. [2]

   • A1 for \(r_s=0.994\), a strong positive correlation
   • R1 for in context: the older the toddler, the more words he or she tends to say

4. (d) Find Pearson's product-moment correlation coefficient, \(r\). [1]

   • A1 for \(r=0.873\)

5. (e) Use the scatter diagram to explain why \(r_s\) is a more suitable measure than \(r\) for these data. [2]

   • R1 for the points follow a curve that always rises (monotonic) but is not a straight line
   • R1 for \(r\) measures only linear correlation, while \(r_s\) measures how well the order agrees, so \(r_s\) is the better description

6. (f) One more toddler is added, aged 26 months, who can say 300 words. Find the new value of \(r_s\) and the new value of \(r\). [2]

   • A1 for \(r_s=0.996\)
   • A1 for \(r=0.832\)

7. (g) Comment on how much each of the two coefficients has changed. [2]

   • A1 for \(r_s\) changes by 0.00179 and \(r\) changes by 0.0410
   • R1 for \(r_s\) hardly changes because the new pair is the largest in both variables, so the order of the data is unchanged, while \(r\) depends on the size of the new value

A factory records the number of faulty bulbs in each of 60 boxes. [10]

1. (a) Find the median number of faulty bulbs. [2]

   • M1 for \(n=60\): the median is the mean of the 30th and 31st values, found from the cumulative frequencies \(14, 25, 34, 42, 49, 55, 60\)
   • A1 for 2

2. (b) Find the lower quartile, the upper quartile and the interquartile range. [3]

   • A1 for \(Q_1=1\) (the mean of the 15th and 16th values)
   • A1 for \(Q_3=4\) (the mean of the 45th and 46th values)
   • A1 for IQR \(=4-1=3\)

3. (c) A new box has 9 faulty bulbs. Show that this is an outlier. [2]

   • M1 for \(Q_3+1.5\times IQR=4+1.5\times3=8.5\)
   • R1 for \(9>8.5\), so it is an outlier

4. (d) A second factory records the time, in minutes, taken to repair each of its machines. The smallest value is 4, the second smallest is 17, the lower quartile is 25, the median is 31, the upper quartile is 38, the second largest is 52 and the largest is 73. Determine whether the data set has any outliers, and state where the whiskers of the box plot end. [3]

   • M1 for IQR \(=38-25=13\) and the fences \(25-1.5\times13=5.5\) and \(38+1.5\times13=57.5\)
   • A1 for outliers 4 (below 5.5) and 73 (above 57.5)
   • A1 for whiskers ending at 17 and 52, the smallest and largest values that are not outliers

On ten summer days a kiosk records the midday temperature, \(x\) °C, and the number of ice creams sold, \(y\) (in tens). x: 14, 16, 17, 19, 21, 22, 24, 25, 27, 29 y: 20, 24, 23, 27, 30, 29, 34, 36, 35, 70 [11]

1. (a) Find the lower quartile and the upper quartile of \(y\). Hence show that 70 is an outlier. [3]

   • A1 for \(Q_1=24\) and \(Q_3=35\)
   • M1 for \(Q_3+1.5\times IQR=35+1.5\times11=51.5\)
   • R1 for \(70>51.5\), so it is an outlier

2. (b) Find Pearson's product-moment correlation coefficient, \(r\). For \(n=10\) the critical value of \(r\) is \(0.632\). Is there significant evidence of correlation between \(x\) and \(y\)? [3]

   • A1 for \(r=0.811\)
   • M1 for comparing \(|r|=0.811\) with the critical value \(0.632\)
   • R1 for \(0.811>0.632\), so there is significant evidence of correlation (positive, and strong)

3. (c) A line of best fit is drawn by eye through the mean point and the point \((31, 52)\). Write down the mean point and find the equation of this line. [3]

   • A1 for the mean point \((21.4, 32.8)\)
   • M1 for the gradient \(\dfrac{52-32.8}{31-21.4}=2\)
   • A1 for \(y-32.8=2(x-21.4)\), so \(y=2x-10\)

4. (d) The regression line of \(y\) on \(x\) is \(y=2.325x-16.957\) and the mean of \(x\) is \(21.4\). Find the mean of \(y\). Explain why this line should not be used to estimate \(x\) from a given value of \(y\). [2]

   • A1 for \(\bar y=2.325\times21.4-16.957=32.8\) (the regression line passes through the mean point)
   • R1 for the line of \(y\) on \(x\) makes the vertical distances small, so it estimates \(y\) from \(x\), not \(x\) from \(y\)