Statistics: exam review Exam review: second paper

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]
2. (b) Find the number of members on the family plan in the sample. [2]
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]
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]

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]
2. (b) Write down the mode. [1]
3. (c) Use your calculator to find the mean. [2]
4. (d) Use your calculator to find the population standard deviation. [1]
5. (e) Find the number of customers who bought more than 3 items. [1]

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

1. (a) Find the value of \(x\). [2]
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]
3. (c) The original median is 620 and the original interquartile range is 55. Find the new median and the new interquartile range. [2]

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

1. (a) Find the mean. [1]
2. (b) Show how to find the population standard deviation without using the statistics menu of your calculator. [3]
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]
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]

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]
2. (b) Find the interquartile range. [1]
3. (c) Show that the greatest value, 55, is an outlier. [2]
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]

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]
2. (b) Find the interquartile range for each. [2]
3. (c) Compare the delays on the two routes. Make one comment about the centre and one about the spread. [2]
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]

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

1. (a) Find the total number of leaves in the sample. [1]
2. (b) Estimate the number of leaves with length between 10 cm and 22 cm. [2]
3. (c) Estimate the mean length. [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]
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]

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]
2. (b) Find the number of cars with at most 2 passengers. [1]
3. (c) Write down the mode. [1]

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]
2. (b) Use the graph to estimate the lower quartile and the upper quartile, and find the interquartile range. [3]
3. (c) Estimate the number of calls with call length greater than 70 seconds. [2]
4. (d) The longest 10% of calls are reviewed by a supervisor. Estimate the shortest call length that is reviewed. [2]
5. (e) Complete a grouped frequency table for the data. [2]
6. (f) Hence estimate the mean call length. [3]

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]
2. (b) Find Pearson's product-moment correlation coefficient, \(r\). [2]
3. (c) Describe the correlation between the advertising spend and the sales. [1]
4. (d) Find the equation of the regression line of \(y\) on \(x\). [2]
5. (e) Interpret the gradient and the \(y\)-intercept of the regression line in the context of the question. [3]
6. (f) Use the regression line to estimate the sales in a month with an advertising spend of 10 hundred euros. [2]
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]
8. (h) The manager says that the correlation shows that spending more on advertising causes higher sales. Comment on this claim. [1]

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]
2. (b) Find Spearman's rank correlation coefficient, \(r_s\). [2]
3. (c) Interpret the value of \(r_s\) in the context of the question. [2]
4. (d) Find Pearson's product-moment correlation coefficient, \(r\). [1]
5. (e) Use the scatter diagram to explain why \(r_s\) is a more suitable measure than \(r\) for these data. [2]
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]
7. (g) Comment on how much each of the two coefficients has changed. [2]

A publisher records the number of printing errors on each of 60 pages. [10]

1. (a) Find the median number of printing errors. [2]
2. (b) Find the lower quartile, the upper quartile and the interquartile range. [3]
3. (c) A new page has 8 printing errors. Show that this is an outlier. [2]
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]

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]
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]
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]
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]