Statistics: exam review Exam review: second paper One question at a time

A gym has 800 members: 320 on the standard plan, 240 on the student plan, 160 on the family plan and 80 on the senior plan. A manager needs a sample of 50 members. The sample has members from each plan in proportion to the number on that plan, chosen at random within each plan. [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 members on the family plan in the sample. [2]

   • M1 for \(\dfrac{160}{800}\times 50\)
   • A1 for 10

3. (c) The survey records (i) the number of visits a member makes in one month, and (ii) the time a member spends on one visit. 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) The time of a visit is recorded to the nearest minute as 48 minutes. Write down the interval of values that the actual time can have. [1]

   • A1 for \(47.5\le t<48.5\)

The table shows the number of items bought by each of 50 customers at a self-checkout. The frequency for 2 items is \(k\). [7]

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

   • M1 for \(35+k=50\)
   • A1 for \(k=15\)

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

   • A1 for 2 (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 \(=2.88\)

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

   • A1 for \(\sigma=1.31\)

5. (e) Find the number of customers who bought more than 3 items. [1]

   • A1 for 15

The five numbers \(x-2\), \(9\), \(2x\), \(4\) and \(12\) have a mean of 10. [6]

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

   • M1 for \((x-2)+9+2x+4+12=5\times10\)
   • A1 for \(x=9\)

2. (b) The monthly pay of the staff in a company, in euros, has mean 640 and standard deviation 42. Every pay \(P\) is changed to a new pay \(1.1P+50\). Find the new mean and the new standard deviation. [2]

   • A1 for new mean \(1.1\times640+50=754\)
   • A1 for new standard deviation \(1.1\times42=46.2\) (the constant does not change the spread)

3. (c) The original median is 620 and the original interquartile range is 55. Find the new median and the new interquartile range. [2]

   • A1 for new median \(1.1\times620+50=732\)
   • A1 for new IQR \(1.1\times55=60.5\)

The points scored by a basketball player in 6 games are 11, 15, 16, 13, 18, 17. [7]

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

   • A1 for mean \(=15\)

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

   • M1 for the deviations \(-4, 0, 1, -2, 3, 2\)
   • M1 for the squares summed: \(34\), then \(\dfrac{34}{6}=5.67\)
   • A1 for \(\sigma=\sqrt{5.67}=2.38\)

3. (c) A second player has a standard deviation of 1.4 points in the number of points scored. State which player is more consistent, and give a reason. [1]

   • A1 for the second player, because its standard deviation is smaller (1.4 compared with 2.38)

4. (d) The scores of a third player are approximately normally distributed with mean 20 points and standard deviation 3 points. Use the 68–95–99.7 rule to find the interval that contains approximately 95% of the scores. [2]

   • M1 for using the interval \(\mu\pm2\sigma\), about 95% of the data
   • A1 for 14 to 26

A teacher records the time, in minutes, that each of 11 students spent on a homework task: 23, 28, 19, 31, 25, 22, 27, 20, 24, 26, 55. [8]

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

   • A1 for median \(=25\)
   • A1 for \(Q_1=22\) (median of the lower 5 values)
   • A1 for \(Q_3=28\) (median of the upper 5 values)

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

   • A1 for \(28-22=6\)

3. (c) Show that the greatest value, 55, is an outlier. [2]

   • M1 for \(Q_3+1.5\times\text{IQR}=28+1.5\times6=37\)
   • R1 for 55 \(>37\), 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 \(24.5\) (it was \(27.3\)) and the median \(24.5\) (it was \(25\))
   • R1 for the median changes less than the mean, so the median is not pulled by the outlier and is the better measure

Two bus routes, P and Q, are compared. The box plots show the delays, in minutes, of a sample of buses on each route. [7]

1. (a) Write down the median for Route P. [1]

   • A1 for 13

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

   • A1 for Route P: \(18-8=10\)
   • A1 for Route Q: \(14-7=7\)

3. (c) Compare the delays on the two routes. Make one comment about the centre and one about the spread. [2]

   • A1 for a comparison of the IQRs (10 and 7): Route Q has the smaller spread
   • A1 for a comparison of the medians: Route P 13, Route Q 9

4. (d) A bus on Route Q has a delay of 21 minutes. Determine whether this delay is an outlier for Route Q. Show your working. [2]

   • M1 for the fences \(Q_1-1.5\times\text{IQR}=-3.5\) and \(Q_3+1.5\times\text{IQR}=24.5\)
   • R1 for 21 inside the fences, so it is not an outlier

The histogram shows the lengths of a sample of leaves. [5]

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

   • A1 for 5+13+20+14+6=58

2. (b) Estimate the number of leaves with length between 10 cm and 22 cm. [2]

   • M1 for taking part of the classes at each end in proportion to the width
   • A1 for 42.5

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

   • M1 for using the midpoints 10, 14, 18, 22, 26
   • A1 for mean \(\approx18.2\)

Team S has \(18\) runners, with a mean time of \(46.5\) minutes for a 10 km race. Team T has \(12\) runners, with a mean time of \(52\) minutes. [4]

1. (a) Find the mean time, in minutes, of all 30 runners. [2]

   • M1 for \(18\times46.5+12\times52=1461\), the total of all the times, divided by \(30\)
   • A1 for \(1461\div30=48.7\)

2. (b) One time in team S was entered as \(39\) minutes but should have been \(57\) minutes. Find the correct mean time, in minutes, of all 30 runners. [2]

   • M1 for the total changing by \(57-39=18\), to \(1479\)
   • A1 for \(1479\div30=49.3\)

A car park attendant records the number of passengers in each car that enters. The table shows the frequencies for one morning. The mean number of passengers is \(3.25\). [5]

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

   • M1 for the total frequency \(25+y\) and the total \(83+3y\)
   • M1 for \(\dfrac{83+3y}{25+y}=3.25\)
   • A1 for \(y=7\)

2. (b) Find the number of cars with at most 2 passengers. [1]

   • A1 for \(3+7=10\)

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

   • A1 for 4 (the value with the greatest frequency)

The lengths of 100 calls to a helpline 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 \(50\)
   • A1 for 56 (accept 54 to 58)

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

   • M1 for reading at \(25\) and \(75\)
   • A1 for \(Q_1\approx38.9\)
   • A1 for \(Q_3\approx73.6\), so IQR \(\approx34.7\) (accept answers within 2 of these)

3. (c) Estimate the number of calls with call length greater than 70 seconds. [2]

   • M1 for reading the cumulative frequency at 70: about 70
   • A1 for \(100-70=30\) (accept 27 to 33)

4. (d) The longest 10% of calls are reviewed by a supervisor. Estimate the shortest call length that is reviewed. [2]

   • M1 for reading across from a cumulative frequency of \(90\)
   • A1 for 90 (accept 88 to 92)

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

   • M1 for the differences of the cumulative frequencies
   • A1 for 8, 18, 30, 28, 12, 4

6. (f) Hence estimate the mean call length. [3]

   • M1 for the midpoints 10, 30, 50, 70, 90, 110
   • M1 for \(\dfrac{\sum fx}{100}\)
   • A1 for \(56\)

A shop records its monthly spending on advertising, \(x\) hundred euros, and its sales, \(y\) thousand euros, for ten months. The advertising spends are 2, 3, 5, 6, 8, 9, 11, 12, 14, 15 and the sales are 10.5, 17.4, 19.1, 16.2, 20.6, 21.4, 25.4, 27.7, 24.6, 25.2. [14]

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

   • A1 for \((8.5, 20.8)\)

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

   • M1 for entering the two lists and finding the correlation coefficient
   • A1 for \(r=0.893\)

3. (c) Describe the correlation between the advertising spend and the sales. [1]

   • A1 for strong and positive (\(r=0.893\))

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

   • M1 for using linear regression
   • A1 for \(y=1.03x+12.0\)

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 100 euros of advertising raises the predicted sales by 1.03 thousand euros
   • A1 for the intercept: the predicted sales with no advertising are 12.0 thousand euros
   • A1 for the gradient and the intercept each stated in context, with units

6. (f) Use the regression line to estimate the sales in a month with an advertising spend of 10 hundred euros. [2]

   • M1 for substituting \(x=10\)
   • A1 for 22.4

7. (g) Kofi uses the regression line to estimate the sales in a month with an advertising spend of 25 hundred euros. Write down the estimate and comment on its reliability. [2]

   • A1 for 37.9
   • R1 for unreliable: 25 is outside the range of the data (2 to 15), so this is extrapolation

8. (h) The manager says that the correlation shows that spending more on advertising causes higher sales. 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 the season or special offers)

A coach records the hours of practice per week of eight students: 1, 2, 3, 4, 5, 6, 7, 8, and their scores in a skills test: 12, 38, 38, 50, 58, 58, 66, 70. The scatter diagram shows the data. [14]

1. (a) Write down the ranks of the hours of practice and of the scores, giving tied values their average rank. [3]

   • M1 for ranking each variable from smallest to largest
   • A1 for the ranks of the hours of practice: 1, 2, 3, 4, 5, 6, 7, 8
   • A1 for the ranks of the scores: 1, 2.5, 2.5, 4, 5.5, 5.5, 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 \(r_s=0.988\)

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

   • A1 for \(r_s=0.988\), a strong positive correlation
   • R1 for in context: the more hours a student practises, the higher the score tends to be

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

   • A1 for \(r=0.947\)

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 student is added, who practises for 9 hours and scores 90. Find the new value of \(r_s\) and the new value of \(r\). [2]

   • A1 for \(r_s=0.992\)
   • A1 for \(r=0.962\)

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

   • A1 for \(r_s\) changes by 0.00361 and \(r\) changes by 0.0148
   • 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 publisher records the number of printing errors on each of 60 pages. [10]

1. (a) Find the median number of printing errors. [2]

   • M1 for \(n=60\): the median is the mean of the 30th and 31st values, found from the cumulative frequencies \(4, 13, 27, 39, 49, 56, 60\)
   • A1 for 3

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

   • A1 for \(Q_1=2\) (the mean of the 15th and 16th values)
   • A1 for \(Q_3=4\) (the mean of the 45th and 46th values)
   • A1 for IQR \(=4-2=2\)

3. (c) A new page has 8 printing errors. Show that this is an outlier. [2]

   • M1 for \(Q_3+1.5\times IQR=4+1.5\times2=7\)
   • R1 for \(8>7\), so it is an outlier

4. (d) A second publisher records the time, in minutes, taken to proofread each of its chapters. The smallest value is 5, the second smallest is 21, the lower quartile is 28, the median is 36, the upper quartile is 42, the second largest is 55 and the largest is 68. Determine whether the data set has any outliers, and state where the whiskers of the box plot end. [3]

   • M1 for IQR \(=42-28=14\) and the fences \(28-1.5\times14=7\) and \(42+1.5\times14=63\)
   • A1 for outliers 5 (below 7) and 68 (above 63)
   • A1 for whiskers ending at 21 and 55, the smallest and largest values that are not outliers

Ten parcel delivery drivers record their years of experience, \(x\), and the number of parcels they deliver per hour, \(y\). x: 2, 3, 4, 5, 6, 7, 8, 9, 10, 12 y: 12, 15, 14, 18, 20, 19, 23, 25, 24, 52 [11]

1. (a) Find the lower quartile and the upper quartile of \(y\). Hence show that 52 is an outlier. [3]

   • A1 for \(Q_1=15\) and \(Q_3=24\)
   • M1 for \(Q_3+1.5\times IQR=24+1.5\times9=37.5\)
   • R1 for \(52>37.5\), 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 \(r=0.844\)
   • M1 for comparing \(|r|=0.844\) with the critical value \(0.632\)
   • R1 for \(0.844>0.632\), 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 \((9, 29.4)\). Write down the mean point and find the equation of this line. [3]

   • A1 for the mean point \((6.6, 22.2)\)
   • M1 for the gradient \(\dfrac{29.4-22.2}{9-6.6}=3\)
   • A1 for \(y-22.2=3(x-6.6)\), so \(y=3x+2.4\)

4. (d) The regression line of \(y\) on \(x\) is \(y=2.985x+2.5\) and the mean of \(x\) is \(6.6\). 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 \(\bar y=2.985\times6.6+2.5=22.2\) (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 \(y\)