Statistics: exam review Exam review

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

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]
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 matches in which more than 2 goals were scored. [1]

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

1. (a) Find the value of \(x\). [2]
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]
3. (c) The original median is 18 and the original interquartile range is 4. Find the new median and the new interquartile range. [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]
2. (b) Show how to find the population standard deviation without using the statistics menu of your calculator. [3]
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]
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]

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]
2. (b) Find the interquartile range. [1]
3. (c) Show that the greatest value, 87, 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 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]
2. (b) Find the interquartile range for each. [2]
3. (c) Compare the delivery times of the two couriers. Make one comment about the centre and one about the spread. [2]
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]

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

1. (a) Find the total number of tomatoes in the sample. [1]
2. (b) Estimate the number of tomatoes with mass between 27 g and 38 g. [2]
3. (c) Estimate the mean mass. [2]

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]
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]

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

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]
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 patients with waiting time greater than 45 minutes. [2]
4. (d) The longest 10% of waiting times are reviewed by the manager. Estimate the shortest waiting time that is reviewed. [2]
5. (e) Complete a grouped frequency table for the data. [2]
6. (f) Hence estimate the mean waiting time. [3]

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]
2. (b) Find Pearson's product-moment correlation coefficient, \(r\). [2]
3. (c) Describe the correlation between the number of hours of training and the 5 km time. [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 5 km time of a runner who trains for 5.5 hours per week. [2]
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]
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]

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

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

1. (a) Find the median number of faulty bulbs. [2]
2. (b) Find the lower quartile, the upper quartile and the interquartile range. [3]
3. (c) A new box has 9 faulty bulbs. Show that this is an outlier. [2]
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]

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]
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 \((31, 52)\). Write down the mean point and find the equation of this line. [3]
4. (d) The regression line of \(y\) on \(x\) is \(y=2.325x-16.957\) 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]