Complete frequency table and histogram together

6 The times taken by 57 athletes to run 100 metres are summarised in the following cumulative frequency table.

1. State how many athletes ran 100 metres in a time between 10.5 and 11.0 seconds.
2. Draw a histogram on graph paper to represent the times taken by these athletes to run 100 metres.
3. Calculate estimates of the mean and variance of the times taken by these athletes.

1. Customers in a shop have to queue to pay.

The partially completed table below and partially completed histogram opposite, give information about the time, \(x\) minutes, spent in the queue by each of 112 customers one day. No customer spent less than 1 minute or longer than 8 minutes in the queue.

1. Complete the table.
2. Complete the histogram. Ting decides to model the frequency density for these 112 customers by a curve with equation $$y = \frac { k } { x ^ { 2 } } \quad 1 \leqslant x \leqslant 8$$ where \(k\) is a constant.
3. Find the value of \(k\)
   \includegraphics[max width=\textwidth, alt={}]{6a0b46f8-7a6a-4ed8-8c7a-9772787f155a-07_1584_1189_285_443}
   Only use this grid if you need to redraw your histogram.
   \includegraphics[max width=\textwidth, alt={}, center]{6a0b46f8-7a6a-4ed8-8c7a-9772787f155a-09_1582_1192_367_440}
   \includegraphics[max width=\textwidth, alt={}, center]{6a0b46f8-7a6a-4ed8-8c7a-9772787f155a-09_2267_51_307_36}

1. The partially completed table and partially completed histogram give information about the ages of passengers on an airline.

There were no passengers aged 90 or over. \includegraphics[max width=\textwidth, alt={}, center]{6dfefd72-338f-40be-ac37-aef56bfaccaa-04_1173_1792_721_139}

1. Complete the histogram.
2. Use linear interpolation to estimate the median age. An outlier is defined as a value greater than \(Q _ { 3 } + 1.5 \times\) interquartile range.
   Given that \(Q _ { 1 } = 27.3\) and \(Q _ { 3 } = 58.9\)
3. determine, giving a reason, whether or not the oldest passenger could be considered as an outlier.
   (2)

7. The partially completed table below summarises the times taken by 120 job applicants to complete a task. A histogram is drawn. The bar representing the \(5 < t \leq 7\) has a width of 1 cm and a height of 5 cm .

1. Given that the bar representing the group \(14 < t \leq 18\) has a height of 4 cm , find the frequency of this group.
   (2)
2. Showing your working, estimate the mean time taken by the 120 job applicants.
   (3) The lower quartile of the times is 9.6 minutes and the upper quartile of the times is 15.5 minutes.
   For these data, an outlier is classified as any value greater than \(Q _ { 3 } + 1.5 \times\) IQR .
3. Showing your working, explain whether or not any of the times taken by these 120 job applicants might be classified as outliers.
   (2) Candidates with the fastest \(5 \%\) of times for the task are given interviews.
4. Estimate the time taken by a job applicant, below which they might be given an interview.
   (2)

1. The partially completed histogram and the partially completed table show the time, to the nearest minute, that a random sample of motorists was delayed by roadworks on a stretch of motorway.
   \includegraphics[max width=\textwidth, alt={}, center]{8f3dbcb4-3260-4493-a230-12577b4ed691-04_1227_1465_354_301}

Estimate the percentage of these motorists who were delayed by the roadworks for between 8.5 and 13.5 minutes.